50:
11606:
107:
324:
8035:
5978:, which assigns to each finite set of reals the number of points in the set. This measure space is not Ď-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The Ď-finite measure spaces have some very convenient properties; Ď-finiteness can be compared in this respect to the
7777:
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9798:
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Semifinite measures generalize sigma-finite measures, in such a way that some big theorems of measure theory that hold for sigma-finite but not arbitrary measures can be extended with little modification to hold for semifinite measures. (To-do: add examples of such theorems; cf. the talk page.)
12491:
First printing. There is a later (2017) second printing. Though usually there is little difference between the first and subsequent printings, in this case the second printing not only deletes from page 53 the
Exercises 36, 40, 41, and 42 of Chapter 2 but also offers a (slightly, but still
7644:
5649:
12492:
substantially) different presentation of part (ii) of
Exercise 17.8. (The second printing's presentation of part (ii) of Exercise 17.8 (on the Luther decomposition) agrees with usual presentations, whereas the first printing's presentation provides a fresh perspective.)
2566:
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5347:
12015:
11881:
8030:{\displaystyle \mu _{\text{sf}}=\mu |_{\mu ^{\text{pre}}(\mathbb {R} _{>0})}\cup \{A\in {\cal {A}}:\sup\{\mu (B):B\in {\cal {P}}(A)\}=+\infty \}\times \{+\infty \}\cup \{A\in {\cal {A}}:\sup\{\mu (B):B\in {\cal {P}}(A)\}<+\infty \}\times \{0\}.}
11062:
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9118:
8708:
2264:
776:
4962:
9634:
6260:
917:
4831:
5555:
9005:
8355:
5551:
2691:
2389:
2425:
3456:
2754:
3810:
8596:
7513:
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5023:
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3005:
9509:
4212:
8855:
6594:
6185:
4017:
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4350:
4129:
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11321:
10154:
5502:
3656:
546:
11371:
10704:
3585:
5094:
10859:
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10297:
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10763:
7235:
7051:
6431:
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5842:
2632:
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2108:
665:
4679:
1603:
674:
8174:
3358:
291:. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of
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582:
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1970:
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1553:
2949:
9454:
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11388:
A measure is said to be s-finite if it is a countable sum of finite measures. S-finite measures are more general than sigma-finite ones and have applications in the theory of
10826:
9629:
9596:
8065:
7505:
6652:
6520:
5725:
1135:
10793:
6721:
12357:
11233:{\displaystyle T:L_{\mathbb {F} }^{\infty }(\mu )\to \left(L_{\mathbb {F} }^{1}(\mu )\right)^{*}:g\mapsto T_{g}=\left(\int fgd\mu \right)_{f\in L_{\mathbb {F} }^{1}(\mu )}.}
10051:{\displaystyle T:L_{\mathbb {F} }^{\infty }(\mu )\to \left(L_{\mathbb {F} }^{1}(\mu )\right)^{*}:g\mapsto T_{g}=\left(\int fgd\mu \right)_{f\in L_{\mathbb {F} }^{1}(\mu )}.}
9344:
8274:
11057:
10227:
9875:
6886:
6807:
6774:
6460:
11032:
11010:
10202:
10180:
9850:
9828:
5225:
1362:
1325:
1000:
14103:
11552:
10985:
10572:
10458:
9391:
9242:
9037:
9032:
8623:
7370:
6113:
4569:
4419:
4386:
4285:
1061:
10915:
10502:
10388:
10341:
10251:
9312:
8905:
8429:
7394:
7167:
6983:
6043:
4064:
3857:
3085:
1180:
12107:
10624:
10598:
9268:
9174:
8381:
8237:
8208:
8744:
6371:
3244:
2158:
1507:
1409:
8925:
8449:
6681:
5774:
5754:
5673:
5437:
5417:
4699:
3298:
3212:
1274:
1087:
608:
478:
410:
389:
9793:{\displaystyle \mu _{0-\infty }=(\sup\{\mu (B)-\mu _{\text{sf}}(B):B\in {\cal {P}}(A)\cap \mu _{\text{sf}}^{\text{pre}}(\mathbb {R} _{\geq 0})\})_{A\in {\cal {A}}}.}
6546:
3709:
3503:
1633:
248:
of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in
9414:
7417:
7313:
5374:
4596:
2918:
2749:
2722:
2420:
2135:
1930:
1903:
1480:
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11258:
10958:
10664:
10644:
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9563:
9364:
9288:
9215:
9138:
8616:
8138:
8118:
7478:
7343:
6741:
6614:
6482:
6133:
6086:
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5798:
5397:
4719:
1868:
1020:
974:
458:
345:
839:
12061:
12038:
11438:
For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive
10938:
10525:
10411:
8948:
8472:
7194:
7143:
7010:
6959:
6909:
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5137:
4724:
4619:
3679:
3139:
1740:
1297:
435:
202:
159:
12288:
Federer, Herbert. Geometric measure theory. Die
Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969 xiv+676 pp.
11278:
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6853:
6830:
6019:
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5186:
5157:
5114:
4252:
4232:
4149:
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3183:
3163:
3116:
3044:
1760:
799:
369:
179:
132:
6189:
5942:
13742:
5507:
3363:
14310:
13820:
11522:
additivity. Historically, this definition was used first. It turns out that in general, finitely additive measures are connected with notions such as
14327:
13837:
1825:
5982:
of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.
8953:
1444:
is a generalization of the
Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties.
8279:
2637:
2335:
3954:
7507:
defined in the above theorem. We give some nice, explicit formulas, which some authors may take as definition, for the semifinite part:
5888:
can be decomposed into a countable union of measurable sets of finite measure. Analogously, a set in a measure space is said to have a
14427:
7639:{\displaystyle \mu _{\text{sf}}=(\sup\{\mu (B):B\in {\cal {P}}(A)\cap \mu ^{\text{pre}}(\mathbb {R} _{\geq 0})\})_{A\in {\cal {A}}}.}
14127:
13145:
13004:
5644:{\displaystyle \mu \left(\bigcup _{\alpha \in \lambda }X_{\alpha }\right)=\sum _{\alpha \in \lambda }\mu \left(X_{\alpha }\right).}
3714:
9150:
Measures that are not semifinite are very wild when restricted to certain sets. Every measure is, in a sense, semifinite once its
8524:
5970:
there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the
1975:
12589:
12348:
1206:
4967:
3091:. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called
1828:, known also as the natural volume form on a symplectic manifold, is useful in classical statistical and Hamiltonian mechanics.
13660:
11630:
4471:
2954:
13491:
12885:
12426:
12257:
9459:
3682:
4154:
2561:{\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)~=~\lim _{i\to \infty }\mu (E_{i})=\sup _{i\geq 1}\mu (E_{i}).}
13031:
12618:
Bengio, Yoshua; Lahlou, Salem; Deleu, Tristan; Hu, Edward J.; Tiwari, Mo; Bengio, Emmanuel (2021). "GFlowNet
Foundations".
8792:
7766:{\displaystyle \mu _{\text{sf}}=(\sup\{\mu (A\cap B):B\in \mu ^{\text{pre}}(\mathbb {R} _{\geq 0})\})_{A\in {\cal {A}}}\}.}
6551:
2884:{\displaystyle \mu \left(\bigcap _{i=1}^{\infty }E_{i}\right)=\lim _{i\to \infty }\mu (E_{i})=\inf _{i\geq 1}\mu (E_{i}).}
14259:
13910:
13652:
6142:
4290:
14653:
14185:
13438:
12238:
12194:
12164:
12146:
11491:. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term
9417:
7420:
4069:
3885:
1234:
14322:
13832:
11556:
11283:
10103:
14241:
12920:
12564:
12541:
12504:
12473:
12454:
12334:
12311:
12279:
12216:
7292:
The zero measure is sigma-finite and thus semifinite. In addition, the zero measure is clearly less than or equal to
5468:
3590:
1662:
498:
93:
71:
11326:
10670:
5342:{\displaystyle \sum _{i\in I}r_{i}=\sup \left\lbrace \sum _{i\in J}r_{i}:|J|<\infty ,J\subseteq I\right\rbrace .}
3519:
64:
14479:
13789:
13779:
5168:
Measures are required to be countably additive. However, the condition can be strengthened as follows. For any set
5028:
10831:
8748:
4424:
1193:
various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in
14559:
14486:
13589:
13498:
13262:
10256:
10709:
7199:
7015:
6376:
14469:
14221:
14201:
7243:
7059:
6280:
5803:
2578:
2276:
2054:
619:
17:
13118:
12010:{\displaystyle (\exists A\in \mu ^{\text{pre}}\{+\infty \})(\forall B\subseteq A)(\mu (B)\in \{0,+\infty \}).}
10870:
Localizable measures are a special case of semifinite measures and a generalization of sigma-finite measures.
4624:
1649:
gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the
1558:
14474:
14420:
14317:
14251:
14155:
14038:
13827:
13774:
13668:
13574:
12977:
12970:
11876:{\displaystyle (\forall A\in \mu ^{\text{pre}}\{+\infty \})(\exists B\subseteq A)(0<\mu (B)<+\infty ).}
1842:
Measure theory is used in machine learning. One example is the Flow
Induced Probability Measure in GFlowNet.
1214:
11458:. Observe, however, that complex measure is necessarily of finite variation, hence complex measures include
8143:
3303:
1642:
is a generalization of the
Lebesgue measure to sets with non-integer dimension, in particular, fractal sets.
13693:
13673:
13637:
13561:
13281:
12997:
8477:
8070:
922:
552:
9518:
1935:
804:
14305:
14175:
13815:
13594:
13556:
13508:
12965:
11620:
5442:
1512:
2923:
14269:
14211:
14068:
13720:
13688:
13678:
13599:
13566:
13197:
13106:
11730:
11473:
have been studied extensively. A measure that takes values in the set of self-adjoint projections on a
9423:
7426:
1818:
10798:
9601:
9568:
8043:
7483:
6619:
6487:
5692:
1102:
308:
14274:
14206:
13737:
13642:
13418:
13346:
11479:
11427:
10768:
6688:
2694:
481:
269:
14226:
13727:
9317:
9113:{\displaystyle \mu ={\cal {C}}\times \{0\}\cup ({\cal {A}}\setminus {\cal {C}})\times \{+\infty \}.}
8703:{\displaystyle \mu =\{(\emptyset ,0)\}\cup ({\cal {A}}\setminus \{\emptyset \})\times \{+\infty \}.}
8244:
2259:{\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)\leq \sum _{i=1}^{\infty }\mu (E_{i}).}
208:
is required to be 0. A simple example is a volume (how big an object occupies a space) as a measure.
14413:
14300:
14279:
14216:
13810:
13256:
13187:
11655:
11635:
11564:
11511:
11037:
10207:
9855:
6858:
6779:
6746:
6436:
1821:
for a list of these) or not. Negative values lead to signed measures, see "generalizations" below.
1331:
58:
13123:
12817:
11015:
10993:
10185:
10163:
9833:
9811:
5191:
1345:
1308:
979:
347:: The measure of a countable disjoint union is the same as the sum of all measures of each subset.
14648:
14081:
13903:
13579:
13337:
13297:
12990:
11530:
10963:
10550:
10436:
9369:
9220:
9010:
7348:
7238:
6091:
4538:
4391:
4355:
4257:
1034:
237:
10896:
10483:
10369:
10322:
10232:
9293:
8886:
8410:
7375:
7148:
6964:
6024:
4022:
3815:
3055:
1150:
771:{\displaystyle \mu \left(\bigcup _{k=1}^{\infty }E_{k}\right)=\sum _{k=1}^{\infty }\mu (E_{k}).}
14591:
14491:
14231:
14160:
14053:
14033:
13862:
13762:
13584:
13306:
13152:
12778:
12086:
10603:
10577:
9247:
9153:
8360:
8216:
8187:
2049:
1339:
1194:
75:
12519:
8714:
6326:
4957:{\displaystyle \mu \{x\in X:f(x)\geq t\}=\lim _{t_{n}\uparrow t}\mu \{x\in X:f(x)>t_{n}\}.}
3217:
2140:
1485:
1367:
14584:
14543:
14539:
14464:
14437:
14058:
13423:
13376:
13371:
13366:
13091:
12960:
8910:
8434:
6657:
5759:
5730:
5658:
5422:
5402:
4684:
3256:
3188:
1438:
1250:
1072:
593:
463:
395:
374:
12304:
Real and
Abstract Analysis: A Modern Treatment of the Theory of Functions of a Real Variable
6525:
3688:
3464:
1608:
14611:
14516:
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14007:
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13698:
13606:
13316:
13271:
13113:
13036:
12895:
12739:
12607:. Osmania University, Digital Library Of India. Cambridge University Press. pp. 91â98.
12408:
11670:
11650:
11389:
9396:
7399:
7295:
5991:
5684:
5352:
4574:
3142:
2896:
2727:
2700:
2398:
2113:
1908:
1881:
1681:
1453:
221:
38:
12066:
11886:
11758:
11243:
10943:
10649:
10629:
10530:
10416:
10302:
10061:
9548:
9349:
9273:
9200:
9123:
8601:
8123:
8103:
7463:
7328:
6726:
6599:
6467:
6118:
6071:
5847:
5783:
5382:
4704:
1853:
1005:
959:
443:
330:
8:
14606:
14596:
14549:
14392:
14180:
14063:
14002:
13971:
13715:
13705:
13551:
13515:
13341:
13070:
13027:
11715:
11695:
11690:
11640:
11496:
11484:
11415:
7170:
6986:
6255:{\displaystyle {\cal {P}}(A)\cap \mu ^{\text{pre}}(\mathbb {R} _{>0})\neq \emptyset .}
5777:
1448:
1210:
1144:
31:
14405:
13393:
12043:
12020:
11578:
is a generalization in both directions: it is a finitely additive, signed measure. (Cf.
10920:
10507:
10393:
8930:
8454:
7176:
7125:
6992:
6941:
6891:
6048:
5950:
5119:
4601:
3661:
3121:
1722:
1279:
417:
184:
141:
14628:
14511:
14362:
14264:
14170:
14117:
14043:
13976:
13896:
13867:
13627:
13612:
13311:
13192:
13170:
12619:
12602:
12379:
11611:
11500:
11401:
11263:
10876:
10463:
10349:
10081:
8866:
8390:
7105:
6921:
6838:
6815:
6004:
5871:
5652:
5171:
5142:
5099:
4237:
4217:
4134:
3865:
3168:
3148:
3101:
3029:
2392:
1835:
1814:
1810:
1745:
1716:
1198:
784:
354:
249:
164:
117:
5909:
3685:
and thus they are continuous almost everywhere, relative to the
Lebesgue measure. If
1205:. Radon measures have an alternative definition in terms of linear functionals on the
312:
14601:
14564:
14150:
13784:
13520:
13481:
13476:
13383:
13301:
13086:
13059:
12926:
12916:
12881:
12560:
12537:
12500:
12469:
12450:
12422:
12383:
12330:
12307:
12275:
12253:
12234:
12226:
12212:
12190:
12182:
12160:
12142:
11700:
11660:
11625:
11605:
11423:
7054:
3506:
3459:
1806:
1646:
1639:
1441:
1190:
1183:
261:
111:
5979:
3098:
A measure can be extended to a complete one by considering the Ď-algebra of subsets
323:
14554:
14534:
14352:
14291:
14112:
14012:
13967:
13955:
13801:
13710:
13486:
13471:
13461:
13446:
13413:
13408:
13398:
13276:
13251:
13066:
12877:
12853:
12790:
12556:
12508:
12414:
12369:
12123:
11675:
11645:
11488:
11463:
11383:
5975:
5900:
3021:
1779:
1423:
1328:
1302:
1244:
1218:
1065:
590:
288:
265:
12727:
14569:
14506:
14501:
14496:
13877:
13857:
13632:
13530:
13525:
13503:
13361:
13326:
13246:
13140:
12891:
12326:
12208:
11720:
11710:
11560:
11454:
11411:
11407:
10667:
5904:
1775:
1427:
273:
106:
14165:
13981:
13767:
13622:
13617:
13428:
13403:
13356:
13286:
13266:
13226:
13216:
13013:
12392:
12342:
12267:
11725:
11685:
11568:
11459:
11449:
11444:
10099:
1771:
1024:
296:
280:
257:
12418:
292:
14642:
14573:
14529:
14377:
14372:
14357:
14347:
14048:
13962:
13937:
13872:
13535:
13456:
13451:
13351:
13321:
13291:
13241:
13236:
13231:
13221:
13135:
13054:
12515:
12482:
11705:
11680:
11665:
11474:
11470:
1831:
1791:
1787:
1783:
1767:
1669:
1650:
1411:; and every other measure with these properties extends the Lebesgue measure.
1222:
1202:
1139:
668:
392:
300:
12930:
912:{\displaystyle \mu (E)=\mu (E\cup \varnothing )=\mu (E)+\mu (\varnothing ),}
14134:
13933:
13466:
13388:
13128:
12705:
12703:
12374:
11523:
11439:
4826:{\displaystyle \{x\in X:f(x)\geq t\}=\bigcap _{n}\{x\in X:f(x)>t_{n}\}.}
1795:
1434:
1221:(2004) and a number of other sources. For more details, see the article on
438:
304:
13165:
12807:
12805:
13331:
12910:
12841:
12715:
12529:
12463:
12134:
11735:
5971:
5896:
3007:
which all have infinite
Lebesgue measure, but the intersection is empty.
245:
213:
12700:
14459:
14387:
14078:
13175:
12829:
12802:
11419:
7315:
It can be shown there is a greatest measure with these two properties:
2893:
This property is false without the assumption that at least one of the
284:
12751:
12040:
the subspace measure induced by the subspace sigma-algebra induced by
9000:{\displaystyle {\cal {C}}=\{A\in {\cal {A}}:A{\text{ is countable}}\}}
14382:
14367:
13157:
13101:
13096:
11503:, while signed measures are the linear closure of positive measures.
8350:{\displaystyle (\forall A\in {\cal {A}})(\mu (A)\in \{0,+\infty \}).}
2046:
613:
205:
12937:
12688:
12676:
5546:{\displaystyle \bigcup _{\alpha \in \lambda }X_{\alpha }\in \Sigma }
2686:{\displaystyle E_{1}\supseteq E_{2}\supseteq E_{3}\supseteq \ldots }
2384:{\displaystyle E_{1}\subseteq E_{2}\subseteq E_{3}\subseteq \ldots }
14616:
14454:
14449:
14022:
13991:
13942:
13919:
13182:
13041:
12624:
11579:
11527:
6911:
gives an example of a semifinite measure that is not sigma-finite.)
3048:
1419:
253:
3451:{\displaystyle \mu \{x\in X:f(x)\geq t\}=\mu \{x\in X:f(x)>t\}}
12982:
12654:
12652:
12650:
11514:. This is the same as a measure except that instead of requiring
5945:
11410:
is assumed to be true, it can be proved that not all subsets of
2035:
233:
225:
135:
12768:
12766:
12647:
12323:
Set Theory: The Third Millennium Edition, Revised and Expanded
9598:
defined in the above theorem. Here is an explicit formula for
8212:
that is not the zero measure is not semifinite. (Here, we say
4151:
then we are done, so assume otherwise. Then there is a unique
1834:
is widely used in statistical mechanics, often under the name
1801:
In physics an example of a measure is spatial distribution of
12489:(Fourth ed.). Prentice Hall. p. 342, Exercise 17.8.
5651:
The second condition is equivalent to the statement that the
5376:
to be the supremum of all the sums of finitely many of them.
3805:{\displaystyle \{x\in X:f(x)\geq t\}=X=\{x\in X:f(x)>t\},}
1415:
1097:
13888:
8591:{\displaystyle \mu =(\sum _{x\in A}f(x))_{A\in {\cal {A}}}.}
12763:
11563:. Contents remain useful in certain technical problems in
5159:
is continuous almost everywhere, this completes the proof.
1802:
241:
229:
10098:
is injective. (This result has import in the study of the
5018:{\displaystyle \lim _{t_{n}\uparrow t}F\left(t_{n}\right)}
14435:
10861:
is the measure defined in Theorem 39.1 in Berberian '65.)
6855:
is countable). (Thus, counting measure, on the power set
4598:
be a monotonically non-decreasing sequence converging to
1766:
Other 'named' measures used in various theories include:
12912:
Theory of charges: a study of finitely additive measures
12466:
Theory of Charges: A Study of Finitely Additive Measures
9802:
5892:
if it is a countable union of sets with finite measure.
4526:{\displaystyle F\left(t_{0}\right)=G\left(t_{0}\right).}
3000:{\displaystyle E_{n}=[n,\infty )\subseteq \mathbb {R} ,}
12915:. M. Bhaskara Rao. London: Academic Press. p. 35.
12272:
Real Analysis: Modern Techniques and Their Applications
9504:{\displaystyle \mu =\mu _{\text{sf}}+\mu _{0-\infty }.}
3118:
which differ by a negligible set from a measurable set
4207:{\displaystyle t_{0}\in \{-\infty \}\cup [0,+\infty )}
2634:
are measurable sets that are decreasing (meaning that
2332:
are measurable sets that are increasing (meaning that
14084:
12497:
Integral, Measure, and Derivative: A Unified Approach
12089:
12069:
12046:
12023:
11909:
11889:
11781:
11761:
11533:
11442:
with values in the (signed) real numbers is called a
11329:
11286:
11266:
11246:
11065:
11040:
11018:
10996:
10966:
10946:
10923:
10899:
10879:
10834:
10801:
10771:
10712:
10673:
10652:
10632:
10606:
10580:
10553:
10533:
10510:
10486:
10466:
10439:
10419:
10396:
10372:
10352:
10325:
10305:
10259:
10235:
10210:
10188:
10166:
10106:
10084:
10064:
9883:
9858:
9836:
9814:
9637:
9604:
9571:
9551:
9521:
9462:
9426:
9399:
9372:
9352:
9320:
9296:
9276:
9250:
9223:
9203:
9156:
9126:
9040:
9013:
8956:
8933:
8913:
8889:
8869:
8850:{\displaystyle \mu =\{(\emptyset ,0),(X,+\infty )\}.}
8795:
8751:
8717:
8626:
8604:
8527:
8480:
8457:
8437:
8413:
8393:
8363:
8282:
8247:
8219:
8190:
8146:
8126:
8106:
8073:
8046:
7780:
7653:
7516:
7486:
7466:
7429:
7402:
7378:
7351:
7331:
7298:
7246:
7202:
7179:
7151:
7128:
7108:
7062:
7018:
6995:
6967:
6944:
6924:
6894:
6861:
6841:
6818:
6782:
6749:
6729:
6691:
6660:
6622:
6602:
6554:
6528:
6490:
6470:
6439:
6379:
6329:
6283:
6192:
6145:
6121:
6094:
6074:
6051:
6027:
6007:
5953:
5912:
5874:
5850:
5806:
5786:
5762:
5733:
5695:
5661:
5558:
5510:
5471:
5445:
5425:
5405:
5385:
5355:
5233:
5194:
5174:
5145:
5122:
5102:
5031:
4970:
4839:
4727:
4707:
4687:
4627:
4604:
4577:
4541:
4474:
4427:
4394:
4358:
4293:
4260:
4240:
4220:
4157:
4137:
4072:
4025:
3957:
3888:
3868:
3818:
3717:
3691:
3664:
3593:
3522:
3467:
3366:
3306:
3259:
3220:
3191:
3171:
3151:
3124:
3104:
3058:
3032:
2957:
2926:
2899:
2757:
2730:
2703:
2640:
2581:
2428:
2401:
2338:
2279:
2165:
2143:
2116:
2057:
1978:
1938:
1911:
1884:
1856:
1748:
1725:
1611:
1561:
1515:
1488:
1456:
1370:
1348:
1311:
1282:
1253:
1153:
1105:
1075:
1037:
1008:
982:
962:
925:
842:
807:
787:
677:
622:
596:
555:
501:
466:
446:
420:
398:
377:
357:
333:
187:
167:
144:
120:
12664:
11601:
11559:. All these are linked in one way or another to the
6589:{\displaystyle f^{\text{pre}}(\mathbb {R} _{>0})}
12617:
12536:. Providence, R.I.: American Mathematical Society.
12499:, Richard A. Silverman, trans. Dover Publications.
12394:
Real and Functional Analysis, Part A: Real Analysis
12345:and Louis Narens (1987). "measurement, theory of",
9183:
Real and Functional Analysis, Part A: Real Analysis
6180:{\displaystyle A\in \mu ^{\text{pre}}\{+\infty \},}
4012:{\displaystyle \mu \{x\in X:f(x)\geq t\}=+\infty ,}
2724:is measurable; furthermore, if at least one of the
1742:The measure of a set is 1 if it contains the point
956:If the condition of non-negativity is dropped, and
14097:
12826:, part (a) of the Theorem in Section 245E, p. 182.
12464:K. P. S. Bhaskara Rao and M. Bhaskara Rao (1983),
12101:
12075:
12055:
12032:
12009:
11895:
11875:
11767:
11546:
11365:
11315:
11272:
11252:
11232:
11051:
11026:
11004:
10979:
10952:
10932:
10909:
10885:
10853:
10820:
10787:
10757:
10698:
10658:
10638:
10618:
10592:
10566:
10539:
10519:
10496:
10472:
10452:
10425:
10405:
10382:
10358:
10335:
10311:
10291:
10245:
10221:
10196:
10174:
10148:
10090:
10070:
10050:
9869:
9844:
9822:
9792:
9623:
9590:
9557:
9535:
9503:
9448:
9408:
9385:
9358:
9338:
9306:
9282:
9262:
9236:
9209:
9168:
9132:
9112:
9026:
8999:
8942:
8919:
8899:
8875:
8849:
8782:
8738:
8702:
8610:
8590:
8513:
8466:
8443:
8423:
8399:
8375:
8349:
8268:
8231:
8202:
8168:
8132:
8112:
8092:
8059:
8029:
7765:
7638:
7499:
7472:
7445:
7411:
7388:
7364:
7337:
7307:
7275:
7229:
7188:
7161:
7137:
7114:
7091:
7045:
7004:
6977:
6953:
6930:
6903:
6880:
6847:
6824:
6801:
6768:
6735:
6715:
6675:
6646:
6608:
6588:
6540:
6514:
6476:
6454:
6425:
6365:
6315:
6254:
6179:
6127:
6107:
6080:
6060:
6037:
6013:
5962:
5936:
5880:
5856:
5836:
5792:
5768:
5748:
5719:
5667:
5643:
5545:
5496:
5457:
5431:
5411:
5391:
5368:
5341:
5219:
5180:
5151:
5131:
5108:
5088:
5017:
4956:
4825:
4713:
4693:
4673:
4613:
4590:
4563:
4525:
4460:
4413:
4380:
4345:{\displaystyle \mu \{x\in X:f(x)\geq t\}=+\infty }
4344:
4279:
4246:
4226:
4206:
4143:
4123:
4058:
4011:
3939:
3874:
3851:
3804:
3703:
3673:
3650:
3579:
3497:
3450:
3352:
3292:
3238:
3206:
3177:
3157:
3133:
3110:
3079:
3038:
2999:
2943:
2912:
2883:
2743:
2716:
2685:
2626:
2560:
2414:
2383:
2324:
2258:
2152:
2129:
2102:
2025:
1964:
1924:
1897:
1862:
1754:
1734:
1627:
1597:
1547:
1501:
1474:
1403:
1356:
1319:
1291:
1268:
1174:
1129:
1081:
1055:
1014:
994:
968:
946:
911:
836:is met automatically due to countable additivity:
828:
793:
770:
659:
602:
576:
540:
472:
452:
429:
404:
383:
363:
339:
196:
173:
153:
126:
12784:
12733:
12658:
12481:
12447:An Introduction to Integration and Measure Theory
12406:
12390:
4124:{\displaystyle \mu \{x\in X:f(x)>t\}=+\infty }
3940:{\displaystyle \mu \{x\in X:f(x)>t\}=+\infty }
14640:
12876:, Series on Multivariate Analysis, vol. 9,
12601:Heath, T. L. (1897). "Measurement of a Circle".
12128:The Elements of Integration and Lebesgue Measure
11422:, and the non-measurable sets postulated by the
11316:{\displaystyle L_{\mathbb {F} }^{\infty }(\mu )}
10149:{\displaystyle L^{1}=L_{\mathbb {F} }^{1}(\mu )}
9660:
7954:
7859:
7670:
7533:
5985:
5263:
4972:
4883:
3248:
2844:
2806:
2521:
2483:
1569:
279:The intuition behind this concept dates back to
110:Informally, a measure has the property of being
12745:
12301:
5497:{\displaystyle X_{\alpha },\alpha <\lambda }
3651:{\displaystyle G(t):=\mu \{x\in X:f(x)\geq t\}}
1186:is a measure space with a probability measure.
1147:is a measure with total measure oneâââthat is,
541:{\displaystyle E\in \Sigma ,\ \ \mu (E)\geq 0.}
27:Generalization of mass, length, area and volume
11366:{\displaystyle L_{\mathbb {F} }^{1}(\mu )^{*}}
10699:{\displaystyle (\mu \times _{\text{cld}}\nu )}
4287:) and finite to the right. Arguing as above,
3658:are monotonically non-increasing functions of
3580:{\displaystyle F(t):=\mu \{x\in X:f(x)>t\}}
2110:of (not necessarily disjoint) measurable sets
14421:
13904:
12998:
10253:be the topology of convergence in measure on
5089:{\displaystyle F(t)=\mu \{x\in X:f(x)>t\}}
2036:Measure of countable unions and intersections
13743:RieszâMarkovâKakutani representation theorem
12908:
12643:, vol. 2 (Second ed.), p. 221
12468:, London: Academic Press, pp. x + 315,
11998:
11983:
11941:
11932:
11813:
11804:
11495:is used. Positive measures are closed under
10854:{\displaystyle \mu \times _{\text{cld}}\nu }
9764:
9663:
9104:
9095:
9063:
9057:
8994:
8967:
8841:
8802:
8783:{\displaystyle {\cal {A}}=\{\emptyset ,X\},}
8774:
8762:
8730:
8724:
8694:
8685:
8676:
8670:
8651:
8633:
8508:
8493:
8338:
8323:
8263:
8248:
8021:
8015:
8009:
7997:
7957:
7935:
7929:
7920:
7914:
7902:
7862:
7840:
7757:
7734:
7673:
7610:
7536:
6710:
6704:
6171:
6162:
5776:). Nonzero finite measures are analogous to
5083:
5050:
4948:
4908:
4876:
4843:
4817:
4777:
4761:
4728:
4668:
4628:
4461:{\displaystyle F\left(t_{0}\right)=+\infty }
4330:
4297:
4180:
4171:
4109:
4076:
3994:
3961:
3925:
3892:
3796:
3763:
3751:
3718:
3645:
3612:
3574:
3541:
3445:
3412:
3403:
3370:
2026:{\displaystyle \mu (E_{1})\leq \mu (E_{2}).}
637:
623:
12294:Measure Theory, Volume 2: Broad Foundations
12250:Integral, Probability, and Fractal Measures
11755:One way to rephrase our definition is that
11448:, while such a function with values in the
10292:{\displaystyle L_{\mathbb {F} }^{0}(\mu ).}
7372:there exists, among semifinite measures on
5800:is proportional to the probability measure
4621:The monotonically non-increasing sequences
2920:has finite measure. For instance, for each
14428:
14414:
14328:Vitale's random BrunnâMinkowski inequality
13911:
13897:
13838:Vitale's random BrunnâMinkowski inequality
13005:
12991:
12495:Shilov, G. E., and Gurevich, B. L., 1978.
11590:to mean its range its a bounded subset of
10758:{\displaystyle (A\times B)=\mu (A)\nu (B)}
8120:is semifinite. It is also evident that if
7230:{\displaystyle s\in \mathbb {R} _{>0}.}
7046:{\displaystyle s\in \mathbb {R} _{>0}.}
6426:{\displaystyle \mu (A)=\sum _{a\in A}f(a)}
5903:are Ď-finite but not finite. Consider the
2040:
12847:
12623:
12373:
12172:
11336:
11293:
11205:
11113:
11078:
11042:
11020:
10998:
10266:
10212:
10190:
10168:
10126:
10023:
9931:
9896:
9860:
9838:
9816:
9748:
7819:
7718:
7594:
7276:{\displaystyle {\cal {H}}^{s}|{\cal {B}}}
7211:
7092:{\displaystyle {\cal {H}}^{s}|{\cal {B}}}
7027:
6570:
6316:{\displaystyle {\cal {A}}={\cal {P}}(X),}
6274:Every sigma-finite measure is semifinite.
6227:
5837:{\displaystyle {\frac {1}{\mu (X)}}\mu .}
3505:This property is used in connection with
2990:
2934:
2627:{\displaystyle E_{1},E_{2},E_{3},\ldots }
2325:{\displaystyle E_{1},E_{2},E_{3},\ldots }
2103:{\displaystyle E_{1},E_{2},E_{3},\ldots }
1350:
1313:
1239:Some important measures are listed here.
801:has finite measure, then the requirement
660:{\displaystyle \{E_{k}\}_{k=1}^{\infty }}
220:is a generalization and formalization of
94:Learn how and when to remove this message
12302:Hewitt, Edward; Stromberg, Karl (1965).
12202:
12181:
5678:
4674:{\displaystyle \{x\in X:f(x)>t_{n}\}}
3185:is contained in a null set. One defines
2570:
2268:
1598:{\displaystyle {\sqrt {|\det g|}}d^{n}x}
322:
264:. Far-reaching generalizations (such as
181:is less than or equal to the measure of
105:
57:This article includes a list of general
12943:
12859:
12835:
12823:
12811:
12796:
12757:
12721:
12709:
12670:
12638:
12444:
12439:Introduction to Measure and Integration
12349:New Palgrave: A Dictionary of Economics
12291:
12266:
11883:Negating this rephrasing, we find that
10865:
9144:
3095:if every negligible set is measurable.
1207:locally convex topological vector space
14:
14641:
12772:
12553:Measure Theory and Functional Analysis
12550:
12521:Topics in Real and Functional Analysis
12355:
12225:
11395:
8241:to mean a measure whose range lies in
8169:{\displaystyle \mu =\mu _{\text{sf}}.}
6835:semifinite (without regard to whether
5996:
4833:Continuity from above guarantees that
3683:at most countably many discontinuities
3353:{\displaystyle (\Sigma ,{\cal {B}}())}
976:takes on at most one of the values of
14409:
13892:
12986:
12787:, part (b) of Proposition 2.3, p. 90.
12694:
12682:
12600:
12401:The first edition was published with
12247:
12133:
12109:measure that is not the zero measure.
12083:to said subspace sigma-algebra, is a
9803:Results regarding semifinite measures
8514:{\displaystyle f:X\to \{0,+\infty \}}
8383:measures that are not zero measures.
8093:{\displaystyle \mu =\mu _{\text{sf}}}
5780:in the sense that any finite measure
5756:is a finite real number (rather than
947:{\displaystyle \mu (\varnothing )=0.}
577:{\displaystyle \mu (\varnothing )=0.}
14341:Applications & related
13851:Applications & related
12978:Tutorial: Measure Theory for Dummies
12320:
12296:(Hardback ed.). Torres Fremlin.
12154:
11418:; examples of such sets include the
11377:
9536:{\displaystyle \mathbf {0-\infty } }
1965:{\displaystyle E_{1}\subseteq E_{2}}
829:{\displaystyle \mu (\varnothing )=0}
236:) and other common notions, such as
43:
14260:Marcinkiewicz interpolation theorem
12871:
12862:, part (b) of Theorem 243G, p. 159.
12799:, part (a) of Theorem 243G, p. 159.
12748:, part (b) of Example 10.4, p. 127.
12528:
9176:part (the wild part) is taken away.
7287:
7122:be a complete, separable metric on
6938:be a complete, separable metric on
6776:), we see that counting measure on
5458:{\displaystyle \lambda <\kappa }
3087:A subset of a null set is called a
3010:
1845:
1548:{\displaystyle x_{1},\ldots ,x_{n}}
256:, and can be generalized to assume
24:
14186:Symmetric decreasing rearrangement
14090:
13012:
12514:
12407:Mukherjea, A; Pothoven, K (1978).
12391:Mukherjea, A; Pothoven, K (1985).
12096:
11995:
11950:
11938:
11913:
11864:
11822:
11810:
11785:
11586:charges, where we say a charge is
11539:
11433:
11299:
11084:
10969:
10902:
10810:
10780:
10613:
10556:
10489:
10442:
10375:
10328:
10238:
9902:
9780:
9711:
9649:
9616:
9583:
9529:
9493:
9438:
9375:
9299:
9257:
9226:
9163:
9101:
9084:
9074:
9049:
9016:
8978:
8959:
8892:
8835:
8808:
8765:
8754:
8691:
8673:
8662:
8639:
8578:
8521:be not the zero function, and let
8505:
8416:
8370:
8335:
8297:
8286:
8260:
8226:
8197:
8067:is semifinite, it follows that if
8006:
7983:
7946:
7926:
7911:
7888:
7851:
7750:
7626:
7562:
7381:
7354:
7268:
7250:
7154:
7084:
7066:
6970:
6864:
6785:
6752:
6641:
6509:
6354:
6296:
6286:
6246:
6195:
6168:
6097:
6030:
5763:
5705:
5540:
5406:
5349:That is, we define the sum of the
5316:
4688:
4455:
4339:
4198:
4177:
4118:
4003:
3934:
3486:
3480:
3338:
3318:
3310:
3284:
2980:
2944:{\displaystyle n\in \mathbb {N} ,}
2816:
2782:
2493:
2453:
2229:
2190:
2144:
1115:
1076:
1047:
986:
741:
702:
652:
508:
488:if the following conditions hold:
467:
378:
327:Countable additivity of a measure
63:it lacks sufficient corresponding
25:
14665:
12953:
12534:An Introduction to Measure Theory
11903:is not semifinite if and only if
9449:{\displaystyle \mu _{0-\infty }.}
9079:
8667:
7446:{\displaystyle \mu _{\text{sf}}.}
6268:
3948:
1663:list of probability distributions
1189:For measure spaces that are also
932:
900:
870:
814:
562:
30:For the coalgebraic concept, see
13780:Lebesgue differentiation theorem
13661:CarathĂŠodory's extension theorem
12413:(First ed.). Plenum Press.
12397:(Second ed.). Plenum Press.
12157:A Primer of Lebesgue Integration
11631:CarathĂŠodory's extension theorem
11604:
10821:{\displaystyle B\in {\cal {B}}.}
9624:{\displaystyle \mu _{0-\infty }}
9591:{\displaystyle \mu _{0-\infty }}
9526:
9523:
8060:{\displaystyle \mu _{\text{sf}}}
7500:{\displaystyle \mu _{\text{sf}}}
6888:of an arbitrary uncountable set
6647:{\displaystyle f(x)<+\infty }
6515:{\displaystyle f(x)<+\infty }
5720:{\displaystyle (X,\Sigma ,\mu )}
5465:and any family of disjoint sets
1130:{\displaystyle (X,\Sigma ,\mu )}
272:) of measure are widely used in
204:Furthermore, the measure of the
48:
14560:Least-squares spectral analysis
14487:Fundamental theorem of calculus
12909:Bhaskara Rao, K. P. S. (1983).
12902:
12865:
12362:Canadian Journal of Mathematics
12116:
11749:
10788:{\displaystyle A\in {\cal {A}}}
8178:
7480:to mean the semifinite measure
7396:that are less than or equal to
6716:{\displaystyle f=X\times \{1\}}
6484:is sigma-finite if and only if
3015:
1873:
12946:, p. 27, Exercise 1.15.a.
12632:
12611:
12594:
12582:
12233:. Cambridge University Press.
12173:Berberian, Sterling K (1965).
12139:Measure and Integration Theory
12001:
11977:
11971:
11965:
11962:
11947:
11944:
11910:
11867:
11855:
11849:
11837:
11834:
11819:
11816:
11782:
11506:Another generalization is the
11354:
11347:
11310:
11304:
11260:is localizable if and only if
11222:
11216:
11150:
11130:
11124:
11098:
11095:
11089:
10752:
10746:
10740:
10734:
10725:
10713:
10693:
10674:
10666:are semifinite if and only if
10283:
10277:
10143:
10137:
10040:
10034:
9968:
9948:
9942:
9916:
9913:
9907:
9768:
9761:
9743:
9722:
9716:
9697:
9691:
9675:
9669:
9657:
9339:{\displaystyle \mu =\nu +\xi }
9192:Theorem (Luther decomposition)
9181:A. Mukherjea and K. Pothoven,
9089:
9069:
8838:
8823:
8817:
8805:
8679:
8657:
8648:
8636:
8566:
8562:
8556:
8534:
8490:
8341:
8317:
8311:
8305:
8302:
8283:
8269:{\displaystyle \{0,+\infty \}}
7994:
7988:
7969:
7963:
7899:
7893:
7874:
7868:
7832:
7814:
7799:
7738:
7731:
7713:
7691:
7679:
7667:
7614:
7607:
7589:
7573:
7567:
7548:
7542:
7530:
7262:
7078:
6875:
6869:
6796:
6790:
6763:
6757:
6632:
6626:
6583:
6565:
6500:
6494:
6420:
6414:
6389:
6383:
6357:
6342:
6339:
6307:
6301:
6240:
6222:
6206:
6200:
5931:
5913:
5822:
5816:
5743:
5737:
5714:
5696:
5309:
5301:
5074:
5068:
5041:
5035:
4986:
4932:
4926:
4897:
4867:
4861:
4801:
4795:
4752:
4746:
4652:
4646:
4321:
4315:
4201:
4186:
4100:
4094:
4050:
4044:
4035:
4029:
3985:
3979:
3916:
3910:
3843:
3837:
3828:
3822:
3787:
3781:
3742:
3736:
3636:
3630:
3603:
3597:
3565:
3559:
3532:
3526:
3489:
3474:
3436:
3430:
3394:
3388:
3347:
3344:
3341:
3326:
3323:
3307:
3287:
3272:
3269:
3230:
3224:
3201:
3195:
3068:
3062:
2983:
2971:
2875:
2862:
2837:
2824:
2813:
2552:
2539:
2514:
2501:
2490:
2250:
2237:
2017:
2004:
1995:
1982:
1653:). Such a measure is called a
1635:is the usual Lebesgue measure.
1576:
1565:
1469:
1457:
1392:
1389:
1377:
1374:
1263:
1257:
1163:
1157:
1124:
1106:
1050:
1038:
935:
929:
903:
897:
888:
882:
873:
861:
852:
846:
817:
811:
762:
749:
565:
559:
529:
523:
13:
1:
14156:Convergence almost everywhere
13918:
12785:Mukherjea & Pothoven 1985
12734:Royden & Fitzpatrick 2010
12659:Mukherjea & Pothoven 1985
12575:
12358:"A decomposition of measures"
12231:Real Analysis and Probability
12159:, San Diego: Academic Press,
11775:is semifinite if and only if
11469:Measures that take values in
11280:is bijective (if and only if
11052:{\displaystyle \mathbb {C} ,}
10319:is semifinite if and only if
10222:{\displaystyle \mathbb {C} ,}
10078:is semifinite if and only if
9870:{\displaystyle \mathbb {C} ,}
9393:In fact, among such measures
9007:be the countable elements of
6881:{\displaystyle {\cal {P}}(X)}
6802:{\displaystyle {\cal {P}}(X)}
6769:{\displaystyle {\cal {P}}(X)}
6616:is semifinite if and only if
6455:{\displaystyle A\subseteq X.}
5986:Strictly localizable measures
5163:
318:
12485:; Fitzpatrick, P.M. (2010).
12410:Real and Functional Analysis
11027:{\displaystyle \mathbb {R} }
11005:{\displaystyle \mathbb {F} }
10197:{\displaystyle \mathbb {R} }
10175:{\displaystyle \mathbb {F} }
9845:{\displaystyle \mathbb {R} }
9823:{\displaystyle \mathbb {F} }
9346:for some semifinite measure
8357:) Below we give examples of
6812:sigma-finite if and only if
5220:{\displaystyle r_{i},i\in I}
5116:is a point of continuity of
4254:(which can only happen when
1357:{\displaystyle \mathbb {R} }
1320:{\displaystyle \mathbb {R} }
1228:
1217:. This approach is taken by
995:{\displaystyle \pm \infty ,}
7:
14323:PrĂŠkopaâLeindler inequality
14176:Locally integrable function
14098:{\displaystyle L^{\infty }}
13833:PrĂŠkopaâLeindler inequality
12966:Encyclopedia of Mathematics
12746:Hewitt & Stromberg 1965
12403:Part B: Functional Analysis
11621:Abelian von Neumann algebra
11597:
11557:StoneâÄech compactification
11547:{\displaystyle L^{\infty }}
11518:additivity we require only
10980:{\displaystyle {\cal {A}}.}
10567:{\displaystyle {\cal {B}}.}
10453:{\displaystyle {\cal {A}},}
9386:{\displaystyle {\cal {A}}.}
9237:{\displaystyle {\cal {A}},}
9027:{\displaystyle {\cal {A}},}
7365:{\displaystyle {\cal {A}},}
6596:is countable. We have that
6108:{\displaystyle {\cal {A}}.}
5188:and any set of nonnegative
4721:-measurable component, and
4564:{\displaystyle t>t_{0},}
4414:{\displaystyle t_{0}\geq 0}
4381:{\displaystyle t<t_{0}.}
4280:{\displaystyle t_{0}\geq 0}
4234:is infinite to the left of
1809:), or another non-negative
1426:measure is invariant under
1418:measure is invariant under
1197:(and in many cases also in
1056:{\displaystyle (X,\Sigma )}
10:
14670:
14069:Square-integrable function
13775:Lebesgue's density theorem
12874:Random and Vector Measures
12274:(Second ed.). Wiley.
12203:Bourbaki, Nicolas (2004),
11731:Valuation (measure theory)
11462:but not, for example, the
11399:
11381:
10910:{\displaystyle {\cal {A}}}
10497:{\displaystyle {\cal {B}}}
10383:{\displaystyle {\cal {A}}}
10336:{\displaystyle {\cal {T}}}
10246:{\displaystyle {\cal {T}}}
9307:{\displaystyle {\cal {A}}}
8900:{\displaystyle {\cal {A}}}
8424:{\displaystyle {\cal {A}}}
7389:{\displaystyle {\cal {A}}}
7162:{\displaystyle {\cal {B}}}
6978:{\displaystyle {\cal {B}}}
6038:{\displaystyle {\cal {A}}}
5989:
5682:
4701:has at least one finitely
4059:{\displaystyle F(t)=G(t),}
3852:{\displaystyle F(t)=G(t),}
3080:{\displaystyle \mu (X)=0.}
3019:
1509:that in local coordinates
1232:
1175:{\displaystyle \mu (X)=1.}
270:projection-valued measures
36:
29:
14654:Measures (measure theory)
14625:
14525:
14444:
14340:
14318:MinkowskiâSteiner formula
14288:
14250:
14194:
14143:
14077:
14021:
13990:
13926:
13850:
13828:MinkowskiâSteiner formula
13798:
13758:
13751:
13651:
13643:Projection-valued measure
13544:
13437:
13206:
13079:
13020:
12724:, Section 213X, part (c).
12712:, Exercise 11.30, p. 159.
12419:10.1007/978-1-4684-2331-0
12356:Luther, Norman Y (1967).
12248:Edgar, Gerald A. (1998).
12102:{\displaystyle 0-\infty }
11508:finitely additive measure
11480:projection-valued measure
10619:{\displaystyle 0-\infty }
10593:{\displaystyle \mu ,\nu }
9263:{\displaystyle 0-\infty }
9169:{\displaystyle 0-\infty }
8376:{\displaystyle 0-\infty }
8232:{\displaystyle 0-\infty }
8203:{\displaystyle 0-\infty }
7320:Theorem (semifinite part)
1932:are measurable sets with
1235:Measures (measure theory)
482:extended real number line
14301:Isoperimetric inequality
13811:Isoperimetric inequality
13790:VitaliâHahnâSaks theorem
13119:CarathĂŠodory's criterion
12736:, Exercise 17.8, p. 342.
12352:, v. 3, pp. 428â32.
12063:i.e. the restriction of
11742:
11656:Geometric measure theory
11636:Content (measure theory)
11567:; this is the theory of
11565:geometric measure theory
8739:{\displaystyle X=\{0\},}
6366:{\displaystyle f:X\to ,}
3239:{\displaystyle \mu (X).}
2751:has finite measure then
2153:{\displaystyle \Sigma :}
1502:{\displaystyle \mu _{g}}
1482:has a canonical measure
1404:{\displaystyle \mu ()=1}
1338:-algebra containing the
1276:= number of elements in
276:and physics in general.
37:Not to be confused with
14306:BrunnâMinkowski theorem
13816:BrunnâMinkowski theorem
13685:Decomposition theorems
12850:, Theorem 39.1, p. 129.
12838:, Section 245M, p. 188.
12814:, Section 243K, p. 162.
12697:, Theorem 1.5.3, p. 42.
12685:, Theorem 1.5.2, p. 42.
12639:Fremlin, D. H. (2010),
12604:The Works Of Archimedes
12445:Nielsen, Ole A (1997).
12175:Measure and Integration
8920:{\displaystyle \sigma }
8444:{\displaystyle \sigma }
6743:is counting measure on
6676:{\displaystyle x\in X.}
5769:{\displaystyle \infty }
5749:{\displaystyle \mu (X)}
5668:{\displaystyle \kappa }
5432:{\displaystyle \kappa }
5412:{\displaystyle \Sigma }
4694:{\displaystyle \Sigma }
3293:{\displaystyle f:X\to }
3207:{\displaystyle \mu (Y)}
3141:that is, such that the
2041:Countable subadditivity
1269:{\displaystyle \mu (S)}
1082:{\displaystyle \Sigma }
603:{\displaystyle \sigma }
473:{\displaystyle \Sigma }
405:{\displaystyle \sigma }
384:{\displaystyle \Sigma }
309:Constantin CarathĂŠodory
287:tried to calculate the
78:more precise citations.
14492:Calculus of variations
14465:Differential equations
14161:Convergence in measure
14099:
13863:Descriptive set theory
13763:Disintegration theorem
13198:Universally measurable
12760:, Section 211O, p. 15.
12375:10.4153/CJM-1968-092-0
12292:Fremlin, D.H. (2016).
12141:, Berlin: de Gruyter,
12103:
12077:
12057:
12034:
12011:
11897:
11877:
11769:
11582:for information about
11548:
11460:finite signed measures
11367:
11317:
11274:
11254:
11234:
11053:
11028:
11006:
10981:
10954:
10934:
10917:be a sigma-algebra on
10911:
10887:
10855:
10822:
10789:
10759:
10700:
10660:
10640:
10620:
10594:
10568:
10541:
10521:
10504:be a sigma-algebra on
10498:
10474:
10454:
10427:
10407:
10390:be a sigma-algebra on
10384:
10360:
10337:
10313:
10293:
10247:
10223:
10198:
10176:
10150:
10092:
10072:
10052:
9871:
9846:
9824:
9794:
9625:
9592:
9559:
9537:
9505:
9450:
9410:
9387:
9360:
9340:
9308:
9284:
9264:
9238:
9211:
9188:
9170:
9134:
9114:
9028:
9001:
8944:
8921:
8901:
8877:
8851:
8784:
8740:
8704:
8612:
8592:
8515:
8468:
8445:
8425:
8401:
8377:
8351:
8270:
8233:
8204:
8170:
8134:
8114:
8094:
8061:
8031:
7767:
7640:
7501:
7474:
7447:
7413:
7390:
7366:
7339:
7309:
7277:
7231:
7190:
7163:
7139:
7116:
7093:
7047:
7006:
6979:
6955:
6932:
6905:
6882:
6849:
6826:
6803:
6770:
6737:
6717:
6677:
6648:
6610:
6590:
6542:
6541:{\displaystyle x\in X}
6516:
6478:
6456:
6427:
6367:
6317:
6256:
6181:
6129:
6109:
6082:
6062:
6045:be a sigma-algebra on
6039:
6015:
5964:
5938:
5882:
5858:
5838:
5794:
5770:
5750:
5721:
5669:
5645:
5547:
5498:
5459:
5433:
5413:
5393:
5370:
5343:
5221:
5182:
5153:
5133:
5110:
5090:
5019:
4958:
4827:
4715:
4695:
4675:
4615:
4592:
4565:
4527:
4462:
4415:
4382:
4346:
4281:
4248:
4228:
4208:
4145:
4125:
4060:
4013:
3941:
3876:
3853:
3806:
3705:
3704:{\displaystyle t<0}
3675:
3652:
3581:
3499:
3498:{\displaystyle t\in .}
3452:
3354:
3294:
3240:
3208:
3179:
3159:
3135:
3112:
3081:
3040:
3001:
2945:
2914:
2885:
2786:
2745:
2718:
2687:
2628:
2562:
2457:
2416:
2385:
2326:
2260:
2233:
2194:
2154:
2131:
2104:
2027:
1966:
1926:
1899:
1864:
1756:
1736:
1629:
1628:{\displaystyle d^{n}x}
1599:
1549:
1503:
1476:
1405:
1358:
1321:
1293:
1270:
1176:
1131:
1083:
1057:
1016:
996:
970:
948:
913:
830:
795:
772:
745:
706:
661:
604:
578:
542:
474:
454:
431:
406:
385:
365:
348:
341:
209:
198:
175:
155:
128:
14585:Representation theory
14544:quaternionic analysis
14540:Hypercomplex analysis
14438:mathematical analysis
14275:RieszâFischer theorem
14100:
14059:Polarization identity
13665:Convergence theorems
13124:Cylindrical Ď-algebra
12321:Jech, Thomas (2003),
12183:Bogachev, Vladimir I.
12130:, Wiley Interscience.
12104:
12078:
12058:
12035:
12012:
11898:
11878:
11770:
11549:
11428:BanachâTarski paradox
11368:
11318:
11275:
11255:
11235:
11054:
11029:
11007:
10982:
10955:
10935:
10912:
10888:
10856:
10823:
10790:
10760:
10701:
10661:
10641:
10621:
10595:
10569:
10542:
10522:
10499:
10475:
10455:
10428:
10408:
10385:
10361:
10338:
10314:
10294:
10248:
10224:
10199:
10177:
10151:
10093:
10073:
10053:
9872:
9847:
9825:
9795:
9626:
9593:
9560:
9538:
9506:
9451:
9411:
9409:{\displaystyle \xi ,}
9388:
9361:
9341:
9309:
9285:
9265:
9239:
9212:
9171:
9148:
9135:
9120:It can be shown that
9115:
9029:
9002:
8945:
8922:
8902:
8878:
8852:
8785:
8741:
8705:
8613:
8598:It can be shown that
8593:
8516:
8469:
8446:
8426:
8402:
8378:
8352:
8271:
8234:
8205:
8171:
8135:
8115:
8095:
8062:
8032:
7768:
7641:
7502:
7475:
7448:
7414:
7412:{\displaystyle \mu ,}
7391:
7367:
7340:
7310:
7308:{\displaystyle \mu .}
7278:
7232:
7191:
7164:
7140:
7117:
7094:
7048:
7007:
6980:
6956:
6933:
6906:
6883:
6850:
6827:
6804:
6771:
6738:
6718:
6678:
6649:
6611:
6591:
6543:
6517:
6479:
6457:
6428:
6368:
6318:
6257:
6182:
6139:to mean that for all
6130:
6110:
6083:
6063:
6040:
6016:
5965:
5939:
5883:
5859:
5839:
5795:
5771:
5751:
5722:
5679:Sigma-finite measures
5670:
5646:
5548:
5499:
5460:
5439:-additive if for any
5434:
5414:
5394:
5371:
5369:{\displaystyle r_{i}}
5344:
5222:
5183:
5154:
5134:
5111:
5091:
5020:
4959:
4828:
4716:
4696:
4676:
4616:
4593:
4591:{\displaystyle t_{n}}
4566:
4528:
4463:
4416:
4383:
4347:
4282:
4249:
4229:
4209:
4146:
4126:
4061:
4014:
3942:
3877:
3854:
3807:
3706:
3681:so both of them have
3676:
3653:
3582:
3500:
3453:
3355:
3295:
3241:
3209:
3180:
3160:
3136:
3113:
3082:
3041:
3002:
2946:
2915:
2913:{\displaystyle E_{n}}
2886:
2766:
2746:
2744:{\displaystyle E_{n}}
2719:
2717:{\displaystyle E_{n}}
2688:
2629:
2571:Continuity from above
2563:
2437:
2417:
2415:{\displaystyle E_{n}}
2386:
2327:
2269:Continuity from below
2261:
2213:
2174:
2155:
2132:
2130:{\displaystyle E_{n}}
2105:
2028:
1967:
1927:
1925:{\displaystyle E_{2}}
1900:
1898:{\displaystyle E_{1}}
1865:
1757:
1737:
1630:
1600:
1550:
1504:
1477:
1475:{\displaystyle (M,g)}
1406:
1359:
1332:translation-invariant
1322:
1294:
1271:
1177:
1132:
1084:
1069:, and the members of
1058:
1017:
997:
971:
949:
914:
831:
796:
773:
725:
686:
662:
605:
579:
543:
475:
455:
432:
407:
386:
366:
342:
326:
199:
176:
156:
129:
114:in the sense that if
109:
14517:Table of derivatives
14280:RieszâThorin theorem
14123:Infimum and supremum
14082:
14008:Lebesgue integration
13733:Minkowski inequality
13607:Cylinder set measure
13492:Infinite-dimensional
13107:equivalence relation
13037:Lebesgue integration
12590:Measuring the Circle
12551:Weaver, Nik (2013).
12437:M. E. Munroe, 1953.
12405:as a single volume:
12189:, Berlin: Springer,
12087:
12076:{\displaystyle \mu }
12067:
12044:
12021:
11907:
11896:{\displaystyle \mu }
11887:
11779:
11768:{\displaystyle \mu }
11759:
11671:Lebesgue integration
11651:Fuzzy measure theory
11531:
11483:; these are used in
11390:stochastic processes
11327:
11284:
11264:
11253:{\displaystyle \mu }
11244:
11063:
11038:
11016:
10994:
10964:
10953:{\displaystyle \mu }
10944:
10921:
10897:
10877:
10866:Localizable measures
10832:
10799:
10769:
10710:
10671:
10659:{\displaystyle \nu }
10650:
10639:{\displaystyle \mu }
10630:
10604:
10578:
10551:
10540:{\displaystyle \nu }
10531:
10508:
10484:
10464:
10437:
10426:{\displaystyle \mu }
10417:
10394:
10370:
10350:
10323:
10312:{\displaystyle \mu }
10303:
10257:
10233:
10208:
10186:
10164:
10104:
10082:
10071:{\displaystyle \mu }
10062:
9881:
9856:
9834:
9812:
9635:
9602:
9569:
9565:to mean the measure
9558:{\displaystyle \mu }
9549:
9519:
9460:
9424:
9397:
9370:
9359:{\displaystyle \nu }
9350:
9318:
9294:
9283:{\displaystyle \xi }
9274:
9248:
9221:
9210:{\displaystyle \mu }
9201:
9154:
9145:Involved non-example
9133:{\displaystyle \mu }
9124:
9038:
9011:
8954:
8931:
8911:
8887:
8883:be uncountable, let
8867:
8793:
8749:
8715:
8624:
8611:{\displaystyle \mu }
8602:
8525:
8478:
8455:
8435:
8411:
8391:
8361:
8280:
8245:
8217:
8188:
8144:
8133:{\displaystyle \mu }
8124:
8113:{\displaystyle \mu }
8104:
8071:
8044:
7778:
7651:
7514:
7484:
7473:{\displaystyle \mu }
7464:
7427:
7400:
7376:
7349:
7338:{\displaystyle \mu }
7329:
7296:
7244:
7200:
7177:
7149:
7126:
7106:
7060:
7016:
6993:
6965:
6942:
6922:
6892:
6859:
6839:
6816:
6780:
6747:
6736:{\displaystyle \mu }
6727:
6689:
6658:
6620:
6609:{\displaystyle \mu }
6600:
6552:
6526:
6488:
6477:{\displaystyle \mu }
6468:
6437:
6377:
6327:
6281:
6190:
6143:
6128:{\displaystyle \mu }
6119:
6092:
6081:{\displaystyle \mu }
6072:
6049:
6025:
6005:
5992:Decomposable measure
5951:
5910:
5872:
5857:{\displaystyle \mu }
5848:
5804:
5793:{\displaystyle \mu }
5784:
5778:probability measures
5760:
5731:
5727:is called finite if
5693:
5685:Sigma-finite measure
5659:
5556:
5508:
5504:the following hold:
5469:
5443:
5423:
5403:
5392:{\displaystyle \mu }
5383:
5353:
5231:
5192:
5172:
5143:
5120:
5100:
5029:
4968:
4964:The right-hand side
4837:
4725:
4714:{\displaystyle \mu }
4705:
4685:
4625:
4602:
4575:
4539:
4472:
4425:
4392:
4356:
4291:
4258:
4238:
4218:
4155:
4135:
4070:
4023:
3955:
3886:
3866:
3816:
3715:
3689:
3662:
3591:
3520:
3465:
3364:
3304:
3257:
3218:
3189:
3169:
3149:
3143:symmetric difference
3122:
3102:
3056:
3030:
2955:
2924:
2897:
2755:
2728:
2701:
2638:
2579:
2426:
2399:
2336:
2277:
2163:
2141:
2114:
2055:
1976:
1936:
1909:
1882:
1863:{\displaystyle \mu }
1854:
1746:
1723:
1682:Dirac delta function
1609:
1559:
1513:
1486:
1454:
1368:
1346:
1309:
1280:
1251:
1211:continuous functions
1151:
1103:
1073:
1035:
1015:{\displaystyle \mu }
1006:
980:
969:{\displaystyle \mu }
960:
923:
840:
805:
785:
781:If at least one set
675:
620:
594:
587:Countable additivity
553:
499:
464:
453:{\displaystyle \mu }
444:
418:
396:
375:
355:
340:{\displaystyle \mu }
331:
222:geometrical measures
185:
165:
142:
118:
39:Metric (mathematics)
14597:Continuous function
14550:Functional analysis
14242:Young's convolution
14181:Measurable function
14064:Pythagorean theorem
14054:Parseval's identity
14003:Integrable function
13728:HĂślder's inequality
13590:of random variables
13552:Measurable function
13439:Particular measures
13028:Absolute continuity
12872:Rao, M. M. (2012),
12155:Bear, H.S. (2001),
12017:For every such set
11716:Pushforward measure
11696:Measurable function
11691:Measurable cardinal
11497:conical combination
11485:functional analysis
11416:Lebesgue measurable
11396:Non-measurable sets
11346:
11303:
11215:
11123:
11088:
10626:measure, then both
10276:
10136:
10033:
9941:
9906:
9742:
9195: —
8140:is semifinite then
7323: —
7171:Borel sigma-algebra
6987:Borel sigma-algebra
5997:Semifinite measures
3249:"Dropping the Edge"
1655:probability measure
1449:Riemannian manifold
1145:probability measure
656:
216:, the concept of a
32:Measuring coalgebra
14629:Mathematics portal
14512:Lists of integrals
14363:Probability theory
14265:Plancherel theorem
14171:Integral transform
14118:Chebyshev distance
14095:
14044:Euclidean distance
13977:Minkowski distance
13868:Probability theory
13193:Transverse measure
13171:Non-measurable set
13153:Locally measurable
12268:Folland, Gerald B.
12227:Dudley, Richard M.
12099:
12073:
12056:{\displaystyle A,}
12053:
12033:{\displaystyle A,}
12030:
12007:
11893:
11873:
11765:
11612:Mathematics portal
11544:
11510:, also known as a
11501:linear combination
11402:Non-measurable set
11363:
11330:
11313:
11287:
11270:
11250:
11230:
11199:
11107:
11072:
11049:
11024:
11002:
10977:
10950:
10933:{\displaystyle X,}
10930:
10907:
10883:
10851:
10818:
10785:
10755:
10696:
10656:
10636:
10616:
10590:
10564:
10537:
10520:{\displaystyle Y,}
10517:
10494:
10470:
10450:
10423:
10406:{\displaystyle X,}
10403:
10380:
10356:
10333:
10309:
10289:
10260:
10243:
10219:
10194:
10172:
10146:
10120:
10088:
10068:
10048:
10017:
9925:
9890:
9867:
9842:
9820:
9790:
9728:
9621:
9588:
9555:
9533:
9501:
9446:
9406:
9383:
9356:
9336:
9304:
9280:
9260:
9234:
9207:
9193:
9166:
9130:
9110:
9024:
8997:
8991: is countable
8943:{\displaystyle X,}
8940:
8917:
8897:
8873:
8847:
8780:
8736:
8700:
8608:
8588:
8552:
8511:
8467:{\displaystyle X,}
8464:
8441:
8421:
8397:
8373:
8347:
8266:
8229:
8200:
8166:
8130:
8110:
8090:
8057:
8027:
7763:
7636:
7497:
7470:
7443:
7409:
7386:
7362:
7335:
7321:
7305:
7273:
7227:
7189:{\displaystyle d,}
7186:
7159:
7138:{\displaystyle X,}
7135:
7112:
7089:
7043:
7005:{\displaystyle d,}
7002:
6975:
6954:{\displaystyle X,}
6951:
6928:
6904:{\displaystyle X,}
6901:
6878:
6845:
6822:
6799:
6766:
6733:
6713:
6673:
6644:
6606:
6586:
6538:
6512:
6474:
6452:
6423:
6410:
6363:
6313:
6252:
6177:
6125:
6105:
6078:
6061:{\displaystyle X,}
6058:
6035:
6011:
5963:{\displaystyle k;}
5960:
5934:
5899:with the standard
5878:
5854:
5834:
5790:
5766:
5746:
5717:
5665:
5641:
5616:
5582:
5543:
5526:
5494:
5455:
5429:
5409:
5389:
5366:
5339:
5286:
5249:
5217:
5178:
5149:
5132:{\displaystyle F.}
5129:
5106:
5086:
5015:
4993:
4954:
4904:
4823:
4776:
4711:
4691:
4671:
4614:{\displaystyle t.}
4611:
4588:
4561:
4523:
4458:
4411:
4378:
4342:
4277:
4244:
4224:
4204:
4141:
4121:
4056:
4009:
3937:
3872:
3849:
3802:
3701:
3674:{\displaystyle t,}
3671:
3648:
3577:
3514:
3495:
3448:
3360:-measurable, then
3350:
3290:
3236:
3204:
3175:
3155:
3134:{\displaystyle X,}
3131:
3108:
3077:
3036:
2997:
2941:
2910:
2881:
2858:
2820:
2741:
2714:
2683:
2624:
2558:
2535:
2497:
2422:is measurable and
2412:
2381:
2322:
2256:
2150:
2127:
2100:
2023:
1962:
1922:
1895:
1860:
1836:canonical ensemble
1811:extensive property
1805:(see for example,
1752:
1735:{\displaystyle S.}
1732:
1717:indicator function
1625:
1595:
1545:
1499:
1472:
1401:
1354:
1317:
1292:{\displaystyle S.}
1289:
1266:
1199:probability theory
1191:topological spaces
1172:
1127:
1079:
1053:
1012:
992:
966:
944:
909:
826:
791:
768:
657:
636:
600:
574:
538:
470:
450:
430:{\displaystyle X.}
427:
402:
381:
361:
349:
337:
254:integration theory
250:probability theory
210:
197:{\displaystyle B.}
194:
171:
154:{\displaystyle B,}
151:
124:
14636:
14635:
14602:Special functions
14565:Harmonic analysis
14403:
14402:
14336:
14335:
14151:Almost everywhere
13936: &
13886:
13885:
13846:
13845:
13575:almost everywhere
13521:Spherical measure
13419:Strictly positive
13347:Projection-valued
13087:Almost everywhere
13060:Probability space
12887:978-981-4350-81-5
12524:, (lecture notes)
12507:. Emphasizes the
12441:. Addison Wesley.
12428:978-1-4684-2333-4
12259:978-1-4419-3112-2
11929:
11801:
11701:Minkowski content
11661:Hausdorff measure
11626:Almost everywhere
11424:Hausdorff paradox
11378:s-finite measures
11273:{\displaystyle T}
10886:{\displaystyle X}
10845:
10687:
10473:{\displaystyle Y}
10359:{\displaystyle X}
10091:{\displaystyle T}
9740:
9735:
9688:
9476:
9191:
8992:
8876:{\displaystyle X}
8537:
8407:be nonempty, let
8400:{\displaystyle X}
8160:
8087:
8054:
7811:
7788:
7710:
7661:
7586:
7524:
7494:
7437:
7319:
7115:{\displaystyle d}
7055:Hausdorff measure
6931:{\displaystyle d}
6848:{\displaystyle X}
6832:is countable; and
6825:{\displaystyle X}
6562:
6395:
6219:
6159:
6014:{\displaystyle X}
5980:LindelĂśf property
5895:For example, the
5881:{\displaystyle X}
5826:
5601:
5567:
5511:
5271:
5234:
5181:{\displaystyle I}
5152:{\displaystyle F}
5109:{\displaystyle t}
4971:
4882:
4767:
4247:{\displaystyle t}
4227:{\displaystyle F}
4144:{\displaystyle t}
4066:as required. If
3875:{\displaystyle t}
3512:
3507:Lebesgue integral
3178:{\displaystyle Y}
3158:{\displaystyle X}
3111:{\displaystyle Y}
3039:{\displaystyle X}
3026:A measurable set
2843:
2805:
2520:
2482:
2481:
2475:
1826:Liouville measure
1807:gravity potential
1755:{\displaystyle a}
1647:probability space
1640:Hausdorff measure
1580:
1442:topological group
1184:probability space
794:{\displaystyle E}
519:
516:
364:{\displaystyle X}
266:spectral measures
262:electrical charge
174:{\displaystyle A}
127:{\displaystyle A}
104:
103:
96:
16:(Redirected from
14661:
14555:Fourier analysis
14535:Complex analysis
14436:Major topics in
14430:
14423:
14416:
14407:
14406:
14353:Fourier analysis
14311:Milman's reverse
14294:
14292:Lebesgue measure
14286:
14285:
14270:RiemannâLebesgue
14113:Bounded function
14104:
14102:
14101:
14096:
14094:
14093:
14013:Taxicab geometry
13968:Measurable space
13913:
13906:
13899:
13890:
13889:
13821:Milman's reverse
13804:
13802:Lebesgue measure
13756:
13755:
13160:
13146:infimum/supremum
13067:Measurable space
13007:
13000:
12993:
12984:
12983:
12974:
12947:
12941:
12935:
12934:
12906:
12900:
12898:
12878:World Scientific
12869:
12863:
12857:
12851:
12845:
12839:
12833:
12827:
12821:
12815:
12809:
12800:
12794:
12788:
12782:
12776:
12770:
12761:
12755:
12749:
12743:
12737:
12731:
12725:
12719:
12713:
12707:
12698:
12692:
12686:
12680:
12674:
12668:
12662:
12656:
12645:
12644:
12636:
12630:
12629:
12627:
12615:
12609:
12608:
12598:
12592:
12586:
12570:
12557:World Scientific
12547:
12525:
12509:Daniell integral
12490:
12478:
12460:
12432:
12398:
12387:
12377:
12339:
12317:
12298:Second printing.
12297:
12285:
12263:
12244:
12221:
12199:
12178:
12169:
12151:
12124:Robert G. Bartle
12110:
12108:
12106:
12105:
12100:
12082:
12080:
12079:
12074:
12062:
12060:
12059:
12054:
12039:
12037:
12036:
12031:
12016:
12014:
12013:
12008:
11931:
11930:
11927:
11902:
11900:
11899:
11894:
11882:
11880:
11879:
11874:
11803:
11802:
11799:
11774:
11772:
11771:
11766:
11753:
11676:Lebesgue measure
11641:Fubini's theorem
11614:
11609:
11608:
11553:
11551:
11550:
11545:
11543:
11542:
11499:but not general
11493:positive measure
11489:spectral theorem
11464:Lebesgue measure
11384:s-finite measure
11372:
11370:
11369:
11364:
11362:
11361:
11345:
11340:
11339:
11322:
11320:
11319:
11314:
11302:
11297:
11296:
11279:
11277:
11276:
11271:
11259:
11257:
11256:
11251:
11239:
11237:
11236:
11231:
11226:
11225:
11214:
11209:
11208:
11191:
11187:
11162:
11161:
11143:
11142:
11137:
11133:
11122:
11117:
11116:
11087:
11082:
11081:
11058:
11056:
11055:
11050:
11045:
11033:
11031:
11030:
11025:
11023:
11011:
11009:
11008:
11003:
11001:
10986:
10984:
10983:
10978:
10973:
10972:
10960:be a measure on
10959:
10957:
10956:
10951:
10939:
10937:
10936:
10931:
10916:
10914:
10913:
10908:
10906:
10905:
10892:
10890:
10889:
10884:
10860:
10858:
10857:
10852:
10847:
10846:
10843:
10827:
10825:
10824:
10819:
10814:
10813:
10794:
10792:
10791:
10786:
10784:
10783:
10764:
10762:
10761:
10756:
10705:
10703:
10702:
10697:
10689:
10688:
10685:
10665:
10663:
10662:
10657:
10645:
10643:
10642:
10637:
10625:
10623:
10622:
10617:
10599:
10597:
10596:
10591:
10573:
10571:
10570:
10565:
10560:
10559:
10547:be a measure on
10546:
10544:
10543:
10538:
10526:
10524:
10523:
10518:
10503:
10501:
10500:
10495:
10493:
10492:
10479:
10477:
10476:
10471:
10459:
10457:
10456:
10451:
10446:
10445:
10433:be a measure on
10432:
10430:
10429:
10424:
10412:
10410:
10409:
10404:
10389:
10387:
10386:
10381:
10379:
10378:
10365:
10363:
10362:
10357:
10342:
10340:
10339:
10334:
10332:
10331:
10318:
10316:
10315:
10310:
10298:
10296:
10295:
10290:
10275:
10270:
10269:
10252:
10250:
10249:
10244:
10242:
10241:
10228:
10226:
10225:
10220:
10215:
10203:
10201:
10200:
10195:
10193:
10181:
10179:
10178:
10173:
10171:
10155:
10153:
10152:
10147:
10135:
10130:
10129:
10116:
10115:
10097:
10095:
10094:
10089:
10077:
10075:
10074:
10069:
10057:
10055:
10054:
10049:
10044:
10043:
10032:
10027:
10026:
10009:
10005:
9980:
9979:
9961:
9960:
9955:
9951:
9940:
9935:
9934:
9905:
9900:
9899:
9876:
9874:
9873:
9868:
9863:
9851:
9849:
9848:
9843:
9841:
9829:
9827:
9826:
9821:
9819:
9799:
9797:
9796:
9791:
9786:
9785:
9784:
9783:
9760:
9759:
9751:
9741:
9738:
9736:
9733:
9715:
9714:
9690:
9689:
9686:
9653:
9652:
9630:
9628:
9627:
9622:
9620:
9619:
9597:
9595:
9594:
9589:
9587:
9586:
9564:
9562:
9561:
9556:
9542:
9540:
9539:
9534:
9532:
9510:
9508:
9507:
9502:
9497:
9496:
9478:
9477:
9474:
9455:
9453:
9452:
9447:
9442:
9441:
9415:
9413:
9412:
9407:
9392:
9390:
9389:
9384:
9379:
9378:
9365:
9363:
9362:
9357:
9345:
9343:
9342:
9337:
9313:
9311:
9310:
9305:
9303:
9302:
9289:
9287:
9286:
9281:
9269:
9267:
9266:
9261:
9243:
9241:
9240:
9235:
9230:
9229:
9216:
9214:
9213:
9208:
9197:For any measure
9196:
9186:
9175:
9173:
9172:
9167:
9139:
9137:
9136:
9131:
9119:
9117:
9116:
9111:
9088:
9087:
9078:
9077:
9053:
9052:
9033:
9031:
9030:
9025:
9020:
9019:
9006:
9004:
9003:
8998:
8993:
8990:
8982:
8981:
8963:
8962:
8949:
8947:
8946:
8941:
8926:
8924:
8923:
8918:
8906:
8904:
8903:
8898:
8896:
8895:
8882:
8880:
8879:
8874:
8856:
8854:
8853:
8848:
8789:
8787:
8786:
8781:
8758:
8757:
8745:
8743:
8742:
8737:
8709:
8707:
8706:
8701:
8666:
8665:
8617:
8615:
8614:
8609:
8597:
8595:
8594:
8589:
8584:
8583:
8582:
8581:
8551:
8520:
8518:
8517:
8512:
8473:
8471:
8470:
8465:
8450:
8448:
8447:
8442:
8430:
8428:
8427:
8422:
8420:
8419:
8406:
8404:
8403:
8398:
8382:
8380:
8379:
8374:
8356:
8354:
8353:
8348:
8301:
8300:
8275:
8273:
8272:
8267:
8238:
8236:
8235:
8230:
8209:
8207:
8206:
8201:
8175:
8173:
8172:
8167:
8162:
8161:
8158:
8139:
8137:
8136:
8131:
8119:
8117:
8116:
8111:
8099:
8097:
8096:
8091:
8089:
8088:
8085:
8066:
8064:
8063:
8058:
8056:
8055:
8052:
8036:
8034:
8033:
8028:
7987:
7986:
7950:
7949:
7892:
7891:
7855:
7854:
7836:
7835:
7831:
7830:
7822:
7813:
7812:
7809:
7802:
7790:
7789:
7786:
7772:
7770:
7769:
7764:
7756:
7755:
7754:
7753:
7730:
7729:
7721:
7712:
7711:
7708:
7663:
7662:
7659:
7645:
7643:
7642:
7637:
7632:
7631:
7630:
7629:
7606:
7605:
7597:
7588:
7587:
7584:
7566:
7565:
7526:
7525:
7522:
7506:
7504:
7503:
7498:
7496:
7495:
7492:
7479:
7477:
7476:
7471:
7452:
7450:
7449:
7444:
7439:
7438:
7435:
7418:
7416:
7415:
7410:
7395:
7393:
7392:
7387:
7385:
7384:
7371:
7369:
7368:
7363:
7358:
7357:
7344:
7342:
7341:
7336:
7325:For any measure
7324:
7314:
7312:
7311:
7306:
7288:Involved example
7282:
7280:
7279:
7274:
7272:
7271:
7265:
7260:
7259:
7254:
7253:
7236:
7234:
7233:
7228:
7223:
7222:
7214:
7195:
7193:
7192:
7187:
7168:
7166:
7165:
7160:
7158:
7157:
7144:
7142:
7141:
7136:
7121:
7119:
7118:
7113:
7098:
7096:
7095:
7090:
7088:
7087:
7081:
7076:
7075:
7070:
7069:
7052:
7050:
7049:
7044:
7039:
7038:
7030:
7011:
7009:
7008:
7003:
6984:
6982:
6981:
6976:
6974:
6973:
6960:
6958:
6957:
6952:
6937:
6935:
6934:
6929:
6910:
6908:
6907:
6902:
6887:
6885:
6884:
6879:
6868:
6867:
6854:
6852:
6851:
6846:
6831:
6829:
6828:
6823:
6808:
6806:
6805:
6800:
6789:
6788:
6775:
6773:
6772:
6767:
6756:
6755:
6742:
6740:
6739:
6734:
6722:
6720:
6719:
6714:
6682:
6680:
6679:
6674:
6653:
6651:
6650:
6645:
6615:
6613:
6612:
6607:
6595:
6593:
6592:
6587:
6582:
6581:
6573:
6564:
6563:
6560:
6547:
6545:
6544:
6539:
6521:
6519:
6518:
6513:
6483:
6481:
6480:
6475:
6461:
6459:
6458:
6453:
6432:
6430:
6429:
6424:
6409:
6372:
6370:
6369:
6364:
6322:
6320:
6319:
6314:
6300:
6299:
6290:
6289:
6261:
6259:
6258:
6253:
6239:
6238:
6230:
6221:
6220:
6217:
6199:
6198:
6186:
6184:
6183:
6178:
6161:
6160:
6157:
6134:
6132:
6131:
6126:
6114:
6112:
6111:
6106:
6101:
6100:
6088:be a measure on
6087:
6085:
6084:
6079:
6067:
6065:
6064:
6059:
6044:
6042:
6041:
6036:
6034:
6033:
6020:
6018:
6017:
6012:
5976:counting measure
5969:
5967:
5966:
5961:
5943:
5941:
5940:
5937:{\displaystyle }
5935:
5905:closed intervals
5901:Lebesgue measure
5890:Ď-finite measure
5887:
5885:
5884:
5879:
5863:
5861:
5860:
5855:
5843:
5841:
5840:
5835:
5827:
5825:
5808:
5799:
5797:
5796:
5791:
5775:
5773:
5772:
5767:
5755:
5753:
5752:
5747:
5726:
5724:
5723:
5718:
5689:A measure space
5674:
5672:
5671:
5666:
5655:of null sets is
5650:
5648:
5647:
5642:
5637:
5633:
5632:
5615:
5597:
5593:
5592:
5591:
5581:
5552:
5550:
5549:
5544:
5536:
5535:
5525:
5503:
5501:
5500:
5495:
5481:
5480:
5464:
5462:
5461:
5456:
5438:
5436:
5435:
5430:
5418:
5416:
5415:
5410:
5398:
5396:
5395:
5390:
5375:
5373:
5372:
5367:
5365:
5364:
5348:
5346:
5345:
5340:
5335:
5331:
5312:
5304:
5296:
5295:
5285:
5259:
5258:
5248:
5226:
5224:
5223:
5218:
5204:
5203:
5187:
5185:
5184:
5179:
5158:
5156:
5155:
5150:
5138:
5136:
5135:
5130:
5115:
5113:
5112:
5107:
5095:
5093:
5092:
5087:
5024:
5022:
5021:
5016:
5014:
5010:
5009:
4992:
4985:
4984:
4963:
4961:
4960:
4955:
4947:
4946:
4903:
4896:
4895:
4832:
4830:
4829:
4824:
4816:
4815:
4775:
4720:
4718:
4717:
4712:
4700:
4698:
4697:
4692:
4680:
4678:
4677:
4672:
4667:
4666:
4620:
4618:
4617:
4612:
4597:
4595:
4594:
4589:
4587:
4586:
4570:
4568:
4567:
4562:
4557:
4556:
4532:
4530:
4529:
4524:
4519:
4515:
4514:
4495:
4491:
4490:
4467:
4465:
4464:
4459:
4448:
4444:
4443:
4420:
4418:
4417:
4412:
4404:
4403:
4387:
4385:
4384:
4379:
4374:
4373:
4351:
4349:
4348:
4343:
4286:
4284:
4283:
4278:
4270:
4269:
4253:
4251:
4250:
4245:
4233:
4231:
4230:
4225:
4213:
4211:
4210:
4205:
4167:
4166:
4150:
4148:
4147:
4142:
4130:
4128:
4127:
4122:
4065:
4063:
4062:
4057:
4018:
4016:
4015:
4010:
3946:
3944:
3943:
3938:
3881:
3879:
3878:
3873:
3858:
3856:
3855:
3850:
3811:
3809:
3808:
3803:
3710:
3708:
3707:
3702:
3680:
3678:
3677:
3672:
3657:
3655:
3654:
3649:
3586:
3584:
3583:
3578:
3504:
3502:
3501:
3496:
3457:
3455:
3454:
3449:
3359:
3357:
3356:
3351:
3322:
3321:
3299:
3297:
3296:
3291:
3245:
3243:
3242:
3237:
3213:
3211:
3210:
3205:
3184:
3182:
3181:
3176:
3164:
3162:
3161:
3156:
3140:
3138:
3137:
3132:
3117:
3115:
3114:
3109:
3086:
3084:
3083:
3078:
3045:
3043:
3042:
3037:
3022:Complete measure
3011:Other properties
3006:
3004:
3003:
2998:
2993:
2967:
2966:
2950:
2948:
2947:
2942:
2937:
2919:
2917:
2916:
2911:
2909:
2908:
2890:
2888:
2887:
2882:
2874:
2873:
2857:
2836:
2835:
2819:
2801:
2797:
2796:
2795:
2785:
2780:
2750:
2748:
2747:
2742:
2740:
2739:
2723:
2721:
2720:
2715:
2713:
2712:
2692:
2690:
2689:
2684:
2676:
2675:
2663:
2662:
2650:
2649:
2633:
2631:
2630:
2625:
2617:
2616:
2604:
2603:
2591:
2590:
2567:
2565:
2564:
2559:
2551:
2550:
2534:
2513:
2512:
2496:
2479:
2473:
2472:
2468:
2467:
2466:
2456:
2451:
2421:
2419:
2418:
2413:
2411:
2410:
2390:
2388:
2387:
2382:
2374:
2373:
2361:
2360:
2348:
2347:
2331:
2329:
2328:
2323:
2315:
2314:
2302:
2301:
2289:
2288:
2265:
2263:
2262:
2257:
2249:
2248:
2232:
2227:
2209:
2205:
2204:
2203:
2193:
2188:
2159:
2157:
2156:
2151:
2136:
2134:
2133:
2128:
2126:
2125:
2109:
2107:
2106:
2101:
2093:
2092:
2080:
2079:
2067:
2066:
2032:
2030:
2029:
2024:
2016:
2015:
1994:
1993:
1971:
1969:
1968:
1963:
1961:
1960:
1948:
1947:
1931:
1929:
1928:
1923:
1921:
1920:
1904:
1902:
1901:
1896:
1894:
1893:
1869:
1867:
1866:
1861:
1846:Basic properties
1819:conservation law
1780:Gaussian measure
1762:and 0 otherwise.
1761:
1759:
1758:
1753:
1741:
1739:
1738:
1733:
1634:
1632:
1631:
1626:
1621:
1620:
1604:
1602:
1601:
1596:
1591:
1590:
1581:
1579:
1568:
1563:
1554:
1552:
1551:
1546:
1544:
1543:
1525:
1524:
1508:
1506:
1505:
1500:
1498:
1497:
1481:
1479:
1478:
1473:
1424:hyperbolic angle
1410:
1408:
1407:
1402:
1363:
1361:
1360:
1355:
1353:
1326:
1324:
1323:
1318:
1316:
1303:Lebesgue measure
1298:
1296:
1295:
1290:
1275:
1273:
1272:
1267:
1245:counting measure
1181:
1179:
1178:
1173:
1136:
1134:
1133:
1128:
1088:
1086:
1085:
1080:
1066:measurable space
1062:
1060:
1059:
1054:
1021:
1019:
1018:
1013:
1001:
999:
998:
993:
975:
973:
972:
967:
953:
951:
950:
945:
918:
916:
915:
910:
835:
833:
832:
827:
800:
798:
797:
792:
777:
775:
774:
769:
761:
760:
744:
739:
721:
717:
716:
715:
705:
700:
666:
664:
663:
658:
655:
650:
635:
634:
609:
607:
606:
601:
583:
581:
580:
575:
547:
545:
544:
539:
517:
514:
479:
477:
476:
471:
459:
457:
456:
451:
436:
434:
433:
428:
411:
409:
408:
403:
390:
388:
387:
382:
370:
368:
367:
362:
346:
344:
343:
338:
315:, among others.
289:area of a circle
203:
201:
200:
195:
180:
178:
177:
172:
160:
158:
157:
152:
133:
131:
130:
125:
99:
92:
88:
85:
79:
74:this article by
65:inline citations
52:
51:
44:
21:
14669:
14668:
14664:
14663:
14662:
14660:
14659:
14658:
14639:
14638:
14637:
14632:
14621:
14570:P-adic analysis
14521:
14507:Matrix calculus
14502:Tensor calculus
14497:Vector calculus
14460:Differentiation
14440:
14434:
14404:
14399:
14332:
14289:
14284:
14246:
14222:HausdorffâYoung
14202:BabenkoâBeckner
14190:
14139:
14089:
14085:
14083:
14080:
14079:
14073:
14017:
13986:
13982:Sequence spaces
13922:
13917:
13887:
13882:
13878:Spectral theory
13858:Convex analysis
13842:
13799:
13794:
13747:
13647:
13595:in distribution
13540:
13433:
13263:Logarithmically
13202:
13158:
13141:Essential range
13075:
13016:
13011:
12959:
12956:
12951:
12950:
12942:
12938:
12923:
12907:
12903:
12888:
12870:
12866:
12858:
12854:
12846:
12842:
12834:
12830:
12822:
12818:
12810:
12803:
12795:
12791:
12783:
12779:
12771:
12764:
12756:
12752:
12744:
12740:
12732:
12728:
12720:
12716:
12708:
12701:
12693:
12689:
12681:
12677:
12669:
12665:
12657:
12648:
12637:
12633:
12616:
12612:
12599:
12595:
12587:
12583:
12578:
12573:
12567:
12544:
12476:
12457:
12429:
12337:
12327:Springer Verlag
12314:
12282:
12260:
12241:
12219:
12209:Springer Verlag
12197:
12167:
12149:
12119:
12114:
12113:
12088:
12085:
12084:
12068:
12065:
12064:
12045:
12042:
12041:
12022:
12019:
12018:
11926:
11922:
11908:
11905:
11904:
11888:
11885:
11884:
11798:
11794:
11780:
11777:
11776:
11760:
11757:
11756:
11754:
11750:
11745:
11740:
11721:Regular measure
11711:Product measure
11610:
11603:
11600:
11569:Banach measures
11561:axiom of choice
11538:
11534:
11532:
11529:
11528:
11455:complex measure
11450:complex numbers
11436:
11434:Generalizations
11412:Euclidean space
11408:axiom of choice
11404:
11398:
11386:
11380:
11357:
11353:
11341:
11335:
11334:
11328:
11325:
11324:
11298:
11292:
11291:
11285:
11282:
11281:
11265:
11262:
11261:
11245:
11242:
11241:
11210:
11204:
11203:
11192:
11171:
11167:
11166:
11157:
11153:
11138:
11118:
11112:
11111:
11106:
11102:
11101:
11083:
11077:
11076:
11064:
11061:
11060:
11041:
11039:
11036:
11035:
11019:
11017:
11014:
11013:
10997:
10995:
10992:
10991:
10968:
10967:
10965:
10962:
10961:
10945:
10942:
10941:
10922:
10919:
10918:
10901:
10900:
10898:
10895:
10894:
10878:
10875:
10874:
10868:
10842:
10838:
10833:
10830:
10829:
10809:
10808:
10800:
10797:
10796:
10779:
10778:
10770:
10767:
10766:
10711:
10708:
10707:
10684:
10680:
10672:
10669:
10668:
10651:
10648:
10647:
10631:
10628:
10627:
10605:
10602:
10601:
10600:are both not a
10579:
10576:
10575:
10555:
10554:
10552:
10549:
10548:
10532:
10529:
10528:
10509:
10506:
10505:
10488:
10487:
10485:
10482:
10481:
10465:
10462:
10461:
10441:
10440:
10438:
10435:
10434:
10418:
10415:
10414:
10395:
10392:
10391:
10374:
10373:
10371:
10368:
10367:
10351:
10348:
10347:
10327:
10326:
10324:
10321:
10320:
10304:
10301:
10300:
10271:
10265:
10264:
10258:
10255:
10254:
10237:
10236:
10234:
10231:
10230:
10211:
10209:
10206:
10205:
10189:
10187:
10184:
10183:
10167:
10165:
10162:
10161:
10131:
10125:
10124:
10111:
10107:
10105:
10102:
10101:
10083:
10080:
10079:
10063:
10060:
10059:
10028:
10022:
10021:
10010:
9989:
9985:
9984:
9975:
9971:
9956:
9936:
9930:
9929:
9924:
9920:
9919:
9901:
9895:
9894:
9882:
9879:
9878:
9859:
9857:
9854:
9853:
9837:
9835:
9832:
9831:
9815:
9813:
9810:
9809:
9805:
9779:
9778:
9771:
9767:
9752:
9747:
9746:
9737:
9732:
9710:
9709:
9685:
9681:
9642:
9638:
9636:
9633:
9632:
9609:
9605:
9603:
9600:
9599:
9576:
9572:
9570:
9567:
9566:
9550:
9547:
9546:
9522:
9520:
9517:
9516:
9512:
9486:
9482:
9473:
9469:
9461:
9458:
9457:
9431:
9427:
9425:
9422:
9421:
9416:there exists a
9398:
9395:
9394:
9374:
9373:
9371:
9368:
9367:
9351:
9348:
9347:
9319:
9316:
9315:
9298:
9297:
9295:
9292:
9291:
9275:
9272:
9271:
9249:
9246:
9245:
9244:there exists a
9225:
9224:
9222:
9219:
9218:
9202:
9199:
9198:
9194:
9187:
9180:
9155:
9152:
9151:
9147:
9125:
9122:
9121:
9083:
9082:
9073:
9072:
9048:
9047:
9039:
9036:
9035:
9015:
9014:
9012:
9009:
9008:
8989:
8977:
8976:
8958:
8957:
8955:
8952:
8951:
8932:
8929:
8928:
8912:
8909:
8908:
8891:
8890:
8888:
8885:
8884:
8868:
8865:
8864:
8794:
8791:
8790:
8753:
8752:
8750:
8747:
8746:
8716:
8713:
8712:
8661:
8660:
8625:
8622:
8621:
8603:
8600:
8599:
8577:
8576:
8569:
8565:
8541:
8526:
8523:
8522:
8479:
8476:
8475:
8456:
8453:
8452:
8436:
8433:
8432:
8415:
8414:
8412:
8409:
8408:
8392:
8389:
8388:
8362:
8359:
8358:
8296:
8295:
8281:
8278:
8277:
8246:
8243:
8242:
8218:
8215:
8214:
8189:
8186:
8185:
8181:
8157:
8153:
8145:
8142:
8141:
8125:
8122:
8121:
8105:
8102:
8101:
8084:
8080:
8072:
8069:
8068:
8051:
8047:
8045:
8042:
8041:
7982:
7981:
7945:
7944:
7887:
7886:
7850:
7849:
7823:
7818:
7817:
7808:
7804:
7803:
7798:
7797:
7785:
7781:
7779:
7776:
7775:
7749:
7748:
7741:
7737:
7722:
7717:
7716:
7707:
7703:
7658:
7654:
7652:
7649:
7648:
7625:
7624:
7617:
7613:
7598:
7593:
7592:
7583:
7579:
7561:
7560:
7521:
7517:
7515:
7512:
7511:
7491:
7487:
7485:
7482:
7481:
7465:
7462:
7461:
7458:semifinite part
7454:
7434:
7430:
7428:
7425:
7424:
7401:
7398:
7397:
7380:
7379:
7377:
7374:
7373:
7353:
7352:
7350:
7347:
7346:
7330:
7327:
7326:
7322:
7297:
7294:
7293:
7290:
7267:
7266:
7261:
7255:
7249:
7248:
7247:
7245:
7242:
7241:
7239:packing measure
7215:
7210:
7209:
7201:
7198:
7197:
7178:
7175:
7174:
7153:
7152:
7150:
7147:
7146:
7127:
7124:
7123:
7107:
7104:
7103:
7083:
7082:
7077:
7071:
7065:
7064:
7063:
7061:
7058:
7057:
7031:
7026:
7025:
7017:
7014:
7013:
6994:
6991:
6990:
6969:
6968:
6966:
6963:
6962:
6943:
6940:
6939:
6923:
6920:
6919:
6893:
6890:
6889:
6863:
6862:
6860:
6857:
6856:
6840:
6837:
6836:
6817:
6814:
6813:
6784:
6783:
6781:
6778:
6777:
6751:
6750:
6748:
6745:
6744:
6728:
6725:
6724:
6723:above (so that
6690:
6687:
6686:
6659:
6656:
6655:
6621:
6618:
6617:
6601:
6598:
6597:
6574:
6569:
6568:
6559:
6555:
6553:
6550:
6549:
6527:
6524:
6523:
6489:
6486:
6485:
6469:
6466:
6465:
6438:
6435:
6434:
6399:
6378:
6375:
6374:
6328:
6325:
6324:
6295:
6294:
6285:
6284:
6282:
6279:
6278:
6271:
6231:
6226:
6225:
6216:
6212:
6194:
6193:
6191:
6188:
6187:
6156:
6152:
6144:
6141:
6140:
6120:
6117:
6116:
6096:
6095:
6093:
6090:
6089:
6073:
6070:
6069:
6050:
6047:
6046:
6029:
6028:
6026:
6023:
6022:
6006:
6003:
6002:
5999:
5994:
5988:
5952:
5949:
5948:
5911:
5908:
5907:
5873:
5870:
5869:
5849:
5846:
5845:
5812:
5807:
5805:
5802:
5801:
5785:
5782:
5781:
5761:
5758:
5757:
5732:
5729:
5728:
5694:
5691:
5690:
5687:
5681:
5660:
5657:
5656:
5628:
5624:
5620:
5605:
5587:
5583:
5571:
5566:
5562:
5557:
5554:
5553:
5531:
5527:
5515:
5509:
5506:
5505:
5476:
5472:
5470:
5467:
5466:
5444:
5441:
5440:
5424:
5421:
5420:
5404:
5401:
5400:
5384:
5381:
5380:
5360:
5356:
5354:
5351:
5350:
5308:
5300:
5291:
5287:
5275:
5270:
5266:
5254:
5250:
5238:
5232:
5229:
5228:
5199:
5195:
5193:
5190:
5189:
5173:
5170:
5169:
5166:
5161:
5144:
5141:
5140:
5121:
5118:
5117:
5101:
5098:
5097:
5030:
5027:
5026:
5005:
5001:
4997:
4980:
4976:
4975:
4969:
4966:
4965:
4942:
4938:
4891:
4887:
4886:
4838:
4835:
4834:
4811:
4807:
4771:
4726:
4723:
4722:
4706:
4703:
4702:
4686:
4683:
4682:
4662:
4658:
4626:
4623:
4622:
4603:
4600:
4599:
4582:
4578:
4576:
4573:
4572:
4552:
4548:
4540:
4537:
4536:
4510:
4506:
4502:
4486:
4482:
4478:
4473:
4470:
4469:
4439:
4435:
4431:
4426:
4423:
4422:
4399:
4395:
4393:
4390:
4389:
4369:
4365:
4357:
4354:
4353:
4292:
4289:
4288:
4265:
4261:
4259:
4256:
4255:
4239:
4236:
4235:
4219:
4216:
4215:
4162:
4158:
4156:
4153:
4152:
4136:
4133:
4132:
4071:
4068:
4067:
4024:
4021:
4020:
3956:
3953:
3952:
3887:
3884:
3883:
3867:
3864:
3863:
3817:
3814:
3813:
3716:
3713:
3712:
3690:
3687:
3686:
3663:
3660:
3659:
3592:
3589:
3588:
3521:
3518:
3517:
3466:
3463:
3462:
3365:
3362:
3361:
3317:
3316:
3305:
3302:
3301:
3258:
3255:
3254:
3251:
3219:
3216:
3215:
3190:
3187:
3186:
3170:
3167:
3166:
3150:
3147:
3146:
3123:
3120:
3119:
3103:
3100:
3099:
3057:
3054:
3053:
3031:
3028:
3027:
3024:
3018:
3013:
2989:
2962:
2958:
2956:
2953:
2952:
2933:
2925:
2922:
2921:
2904:
2900:
2898:
2895:
2894:
2869:
2865:
2847:
2831:
2827:
2809:
2791:
2787:
2781:
2770:
2765:
2761:
2756:
2753:
2752:
2735:
2731:
2729:
2726:
2725:
2708:
2704:
2702:
2699:
2698:
2671:
2667:
2658:
2654:
2645:
2641:
2639:
2636:
2635:
2612:
2608:
2599:
2595:
2586:
2582:
2580:
2577:
2576:
2573:
2546:
2542:
2524:
2508:
2504:
2486:
2462:
2458:
2452:
2441:
2436:
2432:
2427:
2424:
2423:
2406:
2402:
2400:
2397:
2396:
2369:
2365:
2356:
2352:
2343:
2339:
2337:
2334:
2333:
2310:
2306:
2297:
2293:
2284:
2280:
2278:
2275:
2274:
2271:
2244:
2240:
2228:
2217:
2199:
2195:
2189:
2178:
2173:
2169:
2164:
2161:
2160:
2142:
2139:
2138:
2121:
2117:
2115:
2112:
2111:
2088:
2084:
2075:
2071:
2062:
2058:
2056:
2053:
2052:
2043:
2038:
2011:
2007:
1989:
1985:
1977:
1974:
1973:
1956:
1952:
1943:
1939:
1937:
1934:
1933:
1916:
1912:
1910:
1907:
1906:
1889:
1885:
1883:
1880:
1879:
1876:
1855:
1852:
1851:
1848:
1776:ergodic measure
1747:
1744:
1743:
1724:
1721:
1720:
1714:
1705:
1692:
1679:
1616:
1612:
1610:
1607:
1606:
1586:
1582:
1575:
1564:
1562:
1560:
1557:
1556:
1539:
1535:
1520:
1516:
1514:
1511:
1510:
1493:
1489:
1487:
1484:
1483:
1455:
1452:
1451:
1447:Every (pseudo)
1439:locally compact
1428:squeeze mapping
1369:
1366:
1365:
1349:
1347:
1344:
1343:
1312:
1310:
1307:
1306:
1281:
1278:
1277:
1252:
1249:
1248:
1237:
1233:Main category:
1231:
1215:compact support
1152:
1149:
1148:
1104:
1101:
1100:
1091:measurable sets
1074:
1071:
1070:
1036:
1033:
1032:
1007:
1004:
1003:
981:
978:
977:
961:
958:
957:
924:
921:
920:
841:
838:
837:
806:
803:
802:
786:
783:
782:
756:
752:
740:
729:
711:
707:
701:
690:
685:
681:
676:
673:
672:
651:
640:
630:
626:
621:
618:
617:
595:
592:
591:
554:
551:
550:
500:
497:
496:
465:
462:
461:
445:
442:
441:
419:
416:
415:
397:
394:
393:
376:
373:
372:
356:
353:
352:
332:
329:
328:
321:
313:Maurice FrĂŠchet
274:quantum physics
258:negative values
186:
183:
182:
166:
163:
162:
161:the measure of
143:
140:
139:
119:
116:
115:
100:
89:
83:
80:
70:Please help to
69:
53:
49:
42:
35:
28:
23:
22:
15:
12:
11:
5:
14667:
14657:
14656:
14651:
14649:Measure theory
14634:
14633:
14626:
14623:
14622:
14620:
14619:
14614:
14609:
14604:
14599:
14594:
14588:
14587:
14582:
14580:Measure theory
14577:
14574:P-adic numbers
14567:
14562:
14557:
14552:
14547:
14537:
14532:
14526:
14523:
14522:
14520:
14519:
14514:
14509:
14504:
14499:
14494:
14489:
14484:
14483:
14482:
14477:
14472:
14462:
14457:
14445:
14442:
14441:
14433:
14432:
14425:
14418:
14410:
14401:
14400:
14398:
14397:
14396:
14395:
14390:
14380:
14375:
14370:
14365:
14360:
14355:
14350:
14344:
14342:
14338:
14337:
14334:
14333:
14331:
14330:
14325:
14320:
14315:
14314:
14313:
14303:
14297:
14295:
14283:
14282:
14277:
14272:
14267:
14262:
14256:
14254:
14248:
14247:
14245:
14244:
14239:
14234:
14229:
14224:
14219:
14214:
14209:
14204:
14198:
14196:
14192:
14191:
14189:
14188:
14183:
14178:
14173:
14168:
14166:Function space
14163:
14158:
14153:
14147:
14145:
14141:
14140:
14138:
14137:
14132:
14131:
14130:
14120:
14115:
14109:
14107:
14092:
14088:
14075:
14074:
14072:
14071:
14066:
14061:
14056:
14051:
14046:
14041:
14039:CauchyâSchwarz
14036:
14030:
14028:
14019:
14018:
14016:
14015:
14010:
14005:
13999:
13997:
13988:
13987:
13985:
13984:
13979:
13974:
13965:
13960:
13959:
13958:
13948:
13940:
13938:Hilbert spaces
13930:
13928:
13927:Basic concepts
13924:
13923:
13916:
13915:
13908:
13901:
13893:
13884:
13883:
13881:
13880:
13875:
13870:
13865:
13860:
13854:
13852:
13848:
13847:
13844:
13843:
13841:
13840:
13835:
13830:
13825:
13824:
13823:
13813:
13807:
13805:
13796:
13795:
13793:
13792:
13787:
13785:Sard's theorem
13782:
13777:
13772:
13771:
13770:
13768:Lifting theory
13759:
13753:
13749:
13748:
13746:
13745:
13740:
13735:
13730:
13725:
13724:
13723:
13721:FubiniâTonelli
13713:
13708:
13703:
13702:
13701:
13696:
13691:
13683:
13682:
13681:
13676:
13671:
13663:
13657:
13655:
13649:
13648:
13646:
13645:
13640:
13635:
13630:
13625:
13620:
13615:
13609:
13604:
13603:
13602:
13600:in probability
13597:
13587:
13582:
13577:
13571:
13570:
13569:
13564:
13559:
13548:
13546:
13542:
13541:
13539:
13538:
13533:
13528:
13523:
13518:
13513:
13512:
13511:
13501:
13496:
13495:
13494:
13484:
13479:
13474:
13469:
13464:
13459:
13454:
13449:
13443:
13441:
13435:
13434:
13432:
13431:
13426:
13421:
13416:
13411:
13406:
13401:
13396:
13391:
13386:
13381:
13380:
13379:
13374:
13369:
13359:
13354:
13349:
13344:
13334:
13329:
13324:
13319:
13314:
13309:
13307:Locally finite
13304:
13294:
13289:
13284:
13279:
13274:
13269:
13259:
13254:
13249:
13244:
13239:
13234:
13229:
13224:
13219:
13213:
13211:
13204:
13203:
13201:
13200:
13195:
13190:
13185:
13180:
13179:
13178:
13168:
13163:
13155:
13150:
13149:
13148:
13138:
13133:
13132:
13131:
13121:
13116:
13111:
13110:
13109:
13099:
13094:
13089:
13083:
13081:
13077:
13076:
13074:
13073:
13064:
13063:
13062:
13052:
13047:
13039:
13034:
13024:
13022:
13021:Basic concepts
13018:
13017:
13014:Measure theory
13010:
13009:
13002:
12995:
12987:
12981:
12980:
12975:
12955:
12954:External links
12952:
12949:
12948:
12936:
12921:
12901:
12886:
12864:
12852:
12848:Berberian 1965
12840:
12828:
12816:
12801:
12789:
12777:
12762:
12750:
12738:
12726:
12714:
12699:
12687:
12675:
12663:
12646:
12641:Measure Theory
12631:
12610:
12593:
12580:
12579:
12577:
12574:
12572:
12571:
12565:
12548:
12542:
12526:
12516:Teschl, Gerald
12512:
12493:
12479:
12474:
12461:
12455:
12442:
12435:
12434:
12433:
12427:
12388:
12353:
12343:R. Duncan Luce
12340:
12335:
12318:
12312:
12299:
12289:
12286:
12280:
12264:
12258:
12245:
12240:978-0521007542
12239:
12223:
12217:
12200:
12196:978-3540345138
12195:
12187:Measure theory
12179:
12170:
12166:978-0120839711
12165:
12152:
12148:978-3110167191
12147:
12131:
12120:
12118:
12115:
12112:
12111:
12098:
12095:
12092:
12072:
12052:
12049:
12029:
12026:
12006:
12003:
12000:
11997:
11994:
11991:
11988:
11985:
11982:
11979:
11976:
11973:
11970:
11967:
11964:
11961:
11958:
11955:
11952:
11949:
11946:
11943:
11940:
11937:
11934:
11925:
11921:
11918:
11915:
11912:
11892:
11872:
11869:
11866:
11863:
11860:
11857:
11854:
11851:
11848:
11845:
11842:
11839:
11836:
11833:
11830:
11827:
11824:
11821:
11818:
11815:
11812:
11809:
11806:
11797:
11793:
11790:
11787:
11784:
11764:
11747:
11746:
11744:
11741:
11739:
11738:
11733:
11728:
11726:Vector measure
11723:
11718:
11713:
11708:
11703:
11698:
11693:
11688:
11686:Lifting theory
11683:
11678:
11673:
11668:
11663:
11658:
11653:
11648:
11643:
11638:
11633:
11628:
11623:
11617:
11616:
11615:
11599:
11596:
11541:
11537:
11526:, the dual of
11445:signed measure
11435:
11432:
11400:Main article:
11397:
11394:
11382:Main article:
11379:
11376:
11375:
11374:
11360:
11356:
11352:
11349:
11344:
11338:
11333:
11312:
11309:
11306:
11301:
11295:
11290:
11269:
11249:
11229:
11224:
11221:
11218:
11213:
11207:
11202:
11198:
11195:
11190:
11186:
11183:
11180:
11177:
11174:
11170:
11165:
11160:
11156:
11152:
11149:
11146:
11141:
11136:
11132:
11129:
11126:
11121:
11115:
11110:
11105:
11100:
11097:
11094:
11091:
11086:
11080:
11075:
11071:
11068:
11048:
11044:
11022:
11000:
10976:
10971:
10949:
10929:
10926:
10904:
10893:be a set, let
10882:
10867:
10864:
10863:
10862:
10850:
10841:
10837:
10817:
10812:
10807:
10804:
10782:
10777:
10774:
10754:
10751:
10748:
10745:
10742:
10739:
10736:
10733:
10730:
10727:
10724:
10721:
10718:
10715:
10695:
10692:
10683:
10679:
10676:
10655:
10635:
10615:
10612:
10609:
10589:
10586:
10583:
10563:
10558:
10536:
10516:
10513:
10491:
10480:be a set, let
10469:
10449:
10444:
10422:
10402:
10399:
10377:
10366:be a set, let
10355:
10346:(Johnson) Let
10344:
10330:
10308:
10288:
10285:
10282:
10279:
10274:
10268:
10263:
10240:
10218:
10214:
10192:
10170:
10158:
10145:
10142:
10139:
10134:
10128:
10123:
10119:
10114:
10110:
10100:dual space of
10087:
10067:
10047:
10042:
10039:
10036:
10031:
10025:
10020:
10016:
10013:
10008:
10004:
10001:
9998:
9995:
9992:
9988:
9983:
9978:
9974:
9970:
9967:
9964:
9959:
9954:
9950:
9947:
9944:
9939:
9933:
9928:
9923:
9918:
9915:
9912:
9909:
9904:
9898:
9893:
9889:
9886:
9866:
9862:
9840:
9818:
9804:
9801:
9789:
9782:
9777:
9774:
9770:
9766:
9763:
9758:
9755:
9750:
9745:
9731:
9727:
9724:
9721:
9718:
9713:
9708:
9705:
9702:
9699:
9696:
9693:
9684:
9680:
9677:
9674:
9671:
9668:
9665:
9662:
9659:
9656:
9651:
9648:
9645:
9641:
9618:
9615:
9612:
9608:
9585:
9582:
9579:
9575:
9554:
9531:
9528:
9525:
9500:
9495:
9492:
9489:
9485:
9481:
9472:
9468:
9465:
9456:Also, we have
9445:
9440:
9437:
9434:
9430:
9405:
9402:
9382:
9377:
9355:
9335:
9332:
9329:
9326:
9323:
9301:
9279:
9259:
9256:
9253:
9233:
9228:
9206:
9189:
9178:
9165:
9162:
9159:
9146:
9143:
9142:
9141:
9129:
9109:
9106:
9103:
9100:
9097:
9094:
9091:
9086:
9081:
9076:
9071:
9068:
9065:
9062:
9059:
9056:
9051:
9046:
9043:
9023:
9018:
8996:
8988:
8985:
8980:
8975:
8972:
8969:
8966:
8961:
8939:
8936:
8916:
8894:
8872:
8861:
8860:
8859:
8858:
8857:
8846:
8843:
8840:
8837:
8834:
8831:
8828:
8825:
8822:
8819:
8816:
8813:
8810:
8807:
8804:
8801:
8798:
8779:
8776:
8773:
8770:
8767:
8764:
8761:
8756:
8735:
8732:
8729:
8726:
8723:
8720:
8699:
8696:
8693:
8690:
8687:
8684:
8681:
8678:
8675:
8672:
8669:
8664:
8659:
8656:
8653:
8650:
8647:
8644:
8641:
8638:
8635:
8632:
8629:
8618:is a measure.
8607:
8587:
8580:
8575:
8572:
8568:
8564:
8561:
8558:
8555:
8550:
8547:
8544:
8540:
8536:
8533:
8530:
8510:
8507:
8504:
8501:
8498:
8495:
8492:
8489:
8486:
8483:
8463:
8460:
8440:
8418:
8396:
8372:
8369:
8366:
8346:
8343:
8340:
8337:
8334:
8331:
8328:
8325:
8322:
8319:
8316:
8313:
8310:
8307:
8304:
8299:
8294:
8291:
8288:
8285:
8265:
8262:
8259:
8256:
8253:
8250:
8228:
8225:
8222:
8199:
8196:
8193:
8180:
8177:
8165:
8156:
8152:
8149:
8129:
8109:
8083:
8079:
8076:
8050:
8038:
8037:
8026:
8023:
8020:
8017:
8014:
8011:
8008:
8005:
8002:
7999:
7996:
7993:
7990:
7985:
7980:
7977:
7974:
7971:
7968:
7965:
7962:
7959:
7956:
7953:
7948:
7943:
7940:
7937:
7934:
7931:
7928:
7925:
7922:
7919:
7916:
7913:
7910:
7907:
7904:
7901:
7898:
7895:
7890:
7885:
7882:
7879:
7876:
7873:
7870:
7867:
7864:
7861:
7858:
7853:
7848:
7845:
7842:
7839:
7834:
7829:
7826:
7821:
7816:
7807:
7801:
7796:
7793:
7784:
7773:
7762:
7759:
7752:
7747:
7744:
7740:
7736:
7733:
7728:
7725:
7720:
7715:
7706:
7702:
7699:
7696:
7693:
7690:
7687:
7684:
7681:
7678:
7675:
7672:
7669:
7666:
7657:
7646:
7635:
7628:
7623:
7620:
7616:
7612:
7609:
7604:
7601:
7596:
7591:
7582:
7578:
7575:
7572:
7569:
7564:
7559:
7556:
7553:
7550:
7547:
7544:
7541:
7538:
7535:
7532:
7529:
7520:
7490:
7469:
7442:
7433:
7408:
7405:
7383:
7361:
7356:
7334:
7317:
7304:
7301:
7289:
7286:
7285:
7284:
7283:is semifinite.
7270:
7264:
7258:
7252:
7226:
7221:
7218:
7213:
7208:
7205:
7185:
7182:
7156:
7134:
7131:
7111:
7100:
7099:is semifinite.
7086:
7080:
7074:
7068:
7042:
7037:
7034:
7029:
7024:
7021:
7001:
6998:
6972:
6950:
6947:
6927:
6916:
6915:
6914:
6913:
6912:
6900:
6897:
6877:
6874:
6871:
6866:
6844:
6833:
6821:
6798:
6795:
6792:
6787:
6765:
6762:
6759:
6754:
6732:
6712:
6709:
6706:
6703:
6700:
6697:
6694:
6683:
6672:
6669:
6666:
6663:
6643:
6640:
6637:
6634:
6631:
6628:
6625:
6605:
6585:
6580:
6577:
6572:
6567:
6558:
6537:
6534:
6531:
6511:
6508:
6505:
6502:
6499:
6496:
6493:
6473:
6451:
6448:
6445:
6442:
6422:
6419:
6416:
6413:
6408:
6405:
6402:
6398:
6394:
6391:
6388:
6385:
6382:
6362:
6359:
6356:
6353:
6350:
6347:
6344:
6341:
6338:
6335:
6332:
6312:
6309:
6306:
6303:
6298:
6293:
6288:
6275:
6270:
6269:Basic examples
6267:
6251:
6248:
6245:
6242:
6237:
6234:
6229:
6224:
6215:
6211:
6208:
6205:
6202:
6197:
6176:
6173:
6170:
6167:
6164:
6155:
6151:
6148:
6124:
6104:
6099:
6077:
6057:
6054:
6032:
6021:be a set, let
6010:
5998:
5995:
5990:Main article:
5987:
5984:
5959:
5956:
5933:
5930:
5927:
5924:
5921:
5918:
5915:
5877:
5853:
5833:
5830:
5824:
5821:
5818:
5815:
5811:
5789:
5765:
5745:
5742:
5739:
5736:
5716:
5713:
5710:
5707:
5704:
5701:
5698:
5683:Main article:
5680:
5677:
5664:
5640:
5636:
5631:
5627:
5623:
5619:
5614:
5611:
5608:
5604:
5600:
5596:
5590:
5586:
5580:
5577:
5574:
5570:
5565:
5561:
5542:
5539:
5534:
5530:
5524:
5521:
5518:
5514:
5493:
5490:
5487:
5484:
5479:
5475:
5454:
5451:
5448:
5428:
5408:
5388:
5363:
5359:
5338:
5334:
5330:
5327:
5324:
5321:
5318:
5315:
5311:
5307:
5303:
5299:
5294:
5290:
5284:
5281:
5278:
5274:
5269:
5265:
5262:
5257:
5253:
5247:
5244:
5241:
5237:
5216:
5213:
5210:
5207:
5202:
5198:
5177:
5165:
5162:
5148:
5128:
5125:
5105:
5085:
5082:
5079:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5055:
5052:
5049:
5046:
5043:
5040:
5037:
5034:
5013:
5008:
5004:
5000:
4996:
4991:
4988:
4983:
4979:
4974:
4953:
4950:
4945:
4941:
4937:
4934:
4931:
4928:
4925:
4922:
4919:
4916:
4913:
4910:
4907:
4902:
4899:
4894:
4890:
4885:
4881:
4878:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4822:
4819:
4814:
4810:
4806:
4803:
4800:
4797:
4794:
4791:
4788:
4785:
4782:
4779:
4774:
4770:
4766:
4763:
4760:
4757:
4754:
4751:
4748:
4745:
4742:
4739:
4736:
4733:
4730:
4710:
4690:
4681:of members of
4670:
4665:
4661:
4657:
4654:
4651:
4648:
4645:
4642:
4639:
4636:
4633:
4630:
4610:
4607:
4585:
4581:
4560:
4555:
4551:
4547:
4544:
4522:
4518:
4513:
4509:
4505:
4501:
4498:
4494:
4489:
4485:
4481:
4477:
4457:
4454:
4451:
4447:
4442:
4438:
4434:
4430:
4410:
4407:
4402:
4398:
4388:Similarly, if
4377:
4372:
4368:
4364:
4361:
4341:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4317:
4314:
4311:
4308:
4305:
4302:
4299:
4296:
4276:
4273:
4268:
4264:
4243:
4223:
4203:
4200:
4197:
4194:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4165:
4161:
4140:
4120:
4117:
4114:
4111:
4108:
4105:
4102:
4099:
4096:
4093:
4090:
4087:
4084:
4081:
4078:
4075:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4028:
4008:
4005:
4002:
3999:
3996:
3993:
3990:
3987:
3984:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3960:
3936:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3871:
3848:
3845:
3842:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3700:
3697:
3694:
3670:
3667:
3647:
3644:
3641:
3638:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3555:
3552:
3549:
3546:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3511:
3494:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3372:
3369:
3349:
3346:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3320:
3315:
3312:
3309:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3265:
3262:
3250:
3247:
3235:
3232:
3229:
3226:
3223:
3203:
3200:
3197:
3194:
3174:
3154:
3130:
3127:
3107:
3089:negligible set
3076:
3073:
3070:
3067:
3064:
3061:
3035:
3020:Main article:
3017:
3014:
3012:
3009:
2996:
2992:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2965:
2961:
2940:
2936:
2932:
2929:
2907:
2903:
2880:
2877:
2872:
2868:
2864:
2861:
2856:
2853:
2850:
2846:
2842:
2839:
2834:
2830:
2826:
2823:
2818:
2815:
2812:
2808:
2804:
2800:
2794:
2790:
2784:
2779:
2776:
2773:
2769:
2764:
2760:
2738:
2734:
2711:
2707:
2682:
2679:
2674:
2670:
2666:
2661:
2657:
2653:
2648:
2644:
2623:
2620:
2615:
2611:
2607:
2602:
2598:
2594:
2589:
2585:
2572:
2569:
2557:
2554:
2549:
2545:
2541:
2538:
2533:
2530:
2527:
2523:
2519:
2516:
2511:
2507:
2503:
2500:
2495:
2492:
2489:
2485:
2478:
2471:
2465:
2461:
2455:
2450:
2447:
2444:
2440:
2435:
2431:
2409:
2405:
2380:
2377:
2372:
2368:
2364:
2359:
2355:
2351:
2346:
2342:
2321:
2318:
2313:
2309:
2305:
2300:
2296:
2292:
2287:
2283:
2270:
2267:
2255:
2252:
2247:
2243:
2239:
2236:
2231:
2226:
2223:
2220:
2216:
2212:
2208:
2202:
2198:
2192:
2187:
2184:
2181:
2177:
2172:
2168:
2149:
2146:
2124:
2120:
2099:
2096:
2091:
2087:
2083:
2078:
2074:
2070:
2065:
2061:
2042:
2039:
2037:
2034:
2022:
2019:
2014:
2010:
2006:
2003:
2000:
1997:
1992:
1988:
1984:
1981:
1959:
1955:
1951:
1946:
1942:
1919:
1915:
1892:
1888:
1875:
1872:
1870:be a measure.
1859:
1847:
1844:
1840:
1839:
1829:
1772:Jordan measure
1764:
1763:
1751:
1731:
1728:
1710:
1701:
1688:
1684:) is given by
1675:
1666:
1665:for instances.
1643:
1636:
1624:
1619:
1615:
1594:
1589:
1585:
1578:
1574:
1571:
1567:
1542:
1538:
1534:
1531:
1528:
1523:
1519:
1496:
1492:
1471:
1468:
1465:
1462:
1459:
1445:
1431:
1412:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1352:
1315:
1299:
1288:
1285:
1265:
1262:
1259:
1256:
1247:is defined by
1230:
1227:
1223:Radon measures
1203:Radon measures
1171:
1168:
1165:
1162:
1159:
1156:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1078:
1052:
1049:
1046:
1043:
1040:
1025:signed measure
1011:
991:
988:
985:
965:
943:
940:
937:
934:
931:
928:
919:and therefore
908:
905:
902:
899:
896:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
863:
860:
857:
854:
851:
848:
845:
825:
822:
819:
816:
813:
810:
790:
779:
778:
767:
764:
759:
755:
751:
748:
743:
738:
735:
732:
728:
724:
720:
714:
710:
704:
699:
696:
693:
689:
684:
680:
654:
649:
646:
643:
639:
633:
629:
625:
599:
584:
573:
570:
567:
564:
561:
558:
548:
537:
534:
531:
528:
525:
522:
513:
510:
507:
504:
493:Non-negativity
469:
449:
426:
423:
401:
380:
360:
336:
320:
317:
297:Henri Lebesgue
281:ancient Greece
193:
190:
170:
150:
147:
123:
102:
101:
56:
54:
47:
26:
18:Measure (math)
9:
6:
4:
3:
2:
14666:
14655:
14652:
14650:
14647:
14646:
14644:
14631:
14630:
14624:
14618:
14615:
14613:
14610:
14608:
14605:
14603:
14600:
14598:
14595:
14593:
14590:
14589:
14586:
14583:
14581:
14578:
14575:
14571:
14568:
14566:
14563:
14561:
14558:
14556:
14553:
14551:
14548:
14545:
14541:
14538:
14536:
14533:
14531:
14530:Real analysis
14528:
14527:
14524:
14518:
14515:
14513:
14510:
14508:
14505:
14503:
14500:
14498:
14495:
14493:
14490:
14488:
14485:
14481:
14478:
14476:
14473:
14471:
14468:
14467:
14466:
14463:
14461:
14458:
14456:
14452:
14451:
14447:
14446:
14443:
14439:
14431:
14426:
14424:
14419:
14417:
14412:
14411:
14408:
14394:
14391:
14389:
14386:
14385:
14384:
14381:
14379:
14378:Sobolev space
14376:
14374:
14373:Real analysis
14371:
14369:
14366:
14364:
14361:
14359:
14358:Lorentz space
14356:
14354:
14351:
14349:
14348:Bochner space
14346:
14345:
14343:
14339:
14329:
14326:
14324:
14321:
14319:
14316:
14312:
14309:
14308:
14307:
14304:
14302:
14299:
14298:
14296:
14293:
14287:
14281:
14278:
14276:
14273:
14271:
14268:
14266:
14263:
14261:
14258:
14257:
14255:
14253:
14249:
14243:
14240:
14238:
14235:
14233:
14230:
14228:
14225:
14223:
14220:
14218:
14215:
14213:
14210:
14208:
14205:
14203:
14200:
14199:
14197:
14193:
14187:
14184:
14182:
14179:
14177:
14174:
14172:
14169:
14167:
14164:
14162:
14159:
14157:
14154:
14152:
14149:
14148:
14146:
14142:
14136:
14133:
14129:
14126:
14125:
14124:
14121:
14119:
14116:
14114:
14111:
14110:
14108:
14106:
14086:
14076:
14070:
14067:
14065:
14062:
14060:
14057:
14055:
14052:
14050:
14049:Hilbert space
14047:
14045:
14042:
14040:
14037:
14035:
14032:
14031:
14029:
14027:
14025:
14020:
14014:
14011:
14009:
14006:
14004:
14001:
14000:
13998:
13996:
13994:
13989:
13983:
13980:
13978:
13975:
13973:
13969:
13966:
13964:
13963:Measure space
13961:
13957:
13954:
13953:
13952:
13949:
13947:
13945:
13941:
13939:
13935:
13932:
13931:
13929:
13925:
13921:
13914:
13909:
13907:
13902:
13900:
13895:
13894:
13891:
13879:
13876:
13874:
13873:Real analysis
13871:
13869:
13866:
13864:
13861:
13859:
13856:
13855:
13853:
13849:
13839:
13836:
13834:
13831:
13829:
13826:
13822:
13819:
13818:
13817:
13814:
13812:
13809:
13808:
13806:
13803:
13797:
13791:
13788:
13786:
13783:
13781:
13778:
13776:
13773:
13769:
13766:
13765:
13764:
13761:
13760:
13757:
13754:
13752:Other results
13750:
13744:
13741:
13739:
13738:RadonâNikodym
13736:
13734:
13731:
13729:
13726:
13722:
13719:
13718:
13717:
13714:
13712:
13711:Fatou's lemma
13709:
13707:
13704:
13700:
13697:
13695:
13692:
13690:
13687:
13686:
13684:
13680:
13677:
13675:
13672:
13670:
13667:
13666:
13664:
13662:
13659:
13658:
13656:
13654:
13650:
13644:
13641:
13639:
13636:
13634:
13631:
13629:
13626:
13624:
13621:
13619:
13616:
13614:
13610:
13608:
13605:
13601:
13598:
13596:
13593:
13592:
13591:
13588:
13586:
13583:
13581:
13578:
13576:
13573:Convergence:
13572:
13568:
13565:
13563:
13560:
13558:
13555:
13554:
13553:
13550:
13549:
13547:
13543:
13537:
13534:
13532:
13529:
13527:
13524:
13522:
13519:
13517:
13514:
13510:
13507:
13506:
13505:
13502:
13500:
13497:
13493:
13490:
13489:
13488:
13485:
13483:
13480:
13478:
13475:
13473:
13470:
13468:
13465:
13463:
13460:
13458:
13455:
13453:
13450:
13448:
13445:
13444:
13442:
13440:
13436:
13430:
13427:
13425:
13422:
13420:
13417:
13415:
13412:
13410:
13407:
13405:
13402:
13400:
13397:
13395:
13392:
13390:
13387:
13385:
13382:
13378:
13377:Outer regular
13375:
13373:
13372:Inner regular
13370:
13368:
13367:Borel regular
13365:
13364:
13363:
13360:
13358:
13355:
13353:
13350:
13348:
13345:
13343:
13339:
13335:
13333:
13330:
13328:
13325:
13323:
13320:
13318:
13315:
13313:
13310:
13308:
13305:
13303:
13299:
13295:
13293:
13290:
13288:
13285:
13283:
13280:
13278:
13275:
13273:
13270:
13268:
13264:
13260:
13258:
13255:
13253:
13250:
13248:
13245:
13243:
13240:
13238:
13235:
13233:
13230:
13228:
13225:
13223:
13220:
13218:
13215:
13214:
13212:
13210:
13205:
13199:
13196:
13194:
13191:
13189:
13186:
13184:
13181:
13177:
13174:
13173:
13172:
13169:
13167:
13164:
13162:
13156:
13154:
13151:
13147:
13144:
13143:
13142:
13139:
13137:
13134:
13130:
13127:
13126:
13125:
13122:
13120:
13117:
13115:
13112:
13108:
13105:
13104:
13103:
13100:
13098:
13095:
13093:
13090:
13088:
13085:
13084:
13082:
13078:
13072:
13068:
13065:
13061:
13058:
13057:
13056:
13055:Measure space
13053:
13051:
13048:
13046:
13044:
13040:
13038:
13035:
13033:
13029:
13026:
13025:
13023:
13019:
13015:
13008:
13003:
13001:
12996:
12994:
12989:
12988:
12985:
12979:
12976:
12972:
12968:
12967:
12962:
12958:
12957:
12945:
12940:
12932:
12928:
12924:
12922:0-12-095780-9
12918:
12914:
12913:
12905:
12897:
12893:
12889:
12883:
12879:
12875:
12868:
12861:
12856:
12849:
12844:
12837:
12832:
12825:
12820:
12813:
12808:
12806:
12798:
12793:
12786:
12781:
12774:
12769:
12767:
12759:
12754:
12747:
12742:
12735:
12730:
12723:
12718:
12711:
12706:
12704:
12696:
12691:
12684:
12679:
12673:, p. 25.
12672:
12667:
12661:, p. 90.
12660:
12655:
12653:
12651:
12642:
12635:
12626:
12621:
12614:
12606:
12605:
12597:
12591:
12585:
12581:
12568:
12566:9789814508568
12562:
12558:
12554:
12549:
12545:
12543:9780821869192
12539:
12535:
12531:
12527:
12523:
12522:
12517:
12513:
12510:
12506:
12505:0-486-63519-8
12502:
12498:
12494:
12488:
12487:Real Analysis
12484:
12480:
12477:
12475:0-12-095780-9
12471:
12467:
12462:
12458:
12456:0-471-59518-7
12452:
12448:
12443:
12440:
12436:
12430:
12424:
12420:
12416:
12412:
12411:
12404:
12400:
12399:
12396:
12395:
12389:
12385:
12381:
12376:
12371:
12367:
12363:
12359:
12354:
12351:
12350:
12344:
12341:
12338:
12336:3-540-44085-2
12332:
12328:
12324:
12319:
12315:
12313:0-387-90138-8
12309:
12305:
12300:
12295:
12290:
12287:
12283:
12281:0-471-31716-0
12277:
12273:
12269:
12265:
12261:
12255:
12251:
12246:
12242:
12236:
12232:
12228:
12224:
12220:
12218:3-540-41129-1
12214:
12210:
12206:
12205:Integration I
12201:
12198:
12192:
12188:
12184:
12180:
12176:
12171:
12168:
12162:
12158:
12153:
12150:
12144:
12140:
12136:
12132:
12129:
12125:
12122:
12121:
12093:
12090:
12070:
12050:
12047:
12027:
12024:
12004:
11992:
11989:
11986:
11980:
11974:
11968:
11959:
11956:
11953:
11935:
11923:
11919:
11916:
11890:
11870:
11861:
11858:
11852:
11846:
11843:
11840:
11831:
11828:
11825:
11807:
11795:
11791:
11788:
11762:
11752:
11748:
11737:
11734:
11732:
11729:
11727:
11724:
11722:
11719:
11717:
11714:
11712:
11709:
11707:
11706:Outer measure
11704:
11702:
11699:
11697:
11694:
11692:
11689:
11687:
11684:
11682:
11681:Lorentz space
11679:
11677:
11674:
11672:
11669:
11667:
11666:Inner measure
11664:
11662:
11659:
11657:
11654:
11652:
11649:
11647:
11646:Fatou's lemma
11644:
11642:
11639:
11637:
11634:
11632:
11629:
11627:
11624:
11622:
11619:
11618:
11613:
11607:
11602:
11595:
11593:
11589:
11585:
11581:
11577:
11572:
11570:
11566:
11562:
11558:
11554:
11535:
11525:
11524:Banach limits
11521:
11517:
11513:
11509:
11504:
11502:
11498:
11494:
11490:
11486:
11482:
11481:
11476:
11475:Hilbert space
11472:
11471:Banach spaces
11467:
11465:
11461:
11457:
11456:
11451:
11447:
11446:
11441:
11431:
11429:
11425:
11421:
11417:
11413:
11409:
11403:
11393:
11391:
11385:
11358:
11350:
11342:
11331:
11307:
11288:
11267:
11247:
11227:
11219:
11211:
11200:
11196:
11193:
11188:
11184:
11181:
11178:
11175:
11172:
11168:
11163:
11158:
11154:
11147:
11144:
11139:
11134:
11127:
11119:
11108:
11103:
11092:
11073:
11069:
11066:
11046:
10989:
10988:
10987:
10974:
10947:
10927:
10924:
10880:
10871:
10848:
10839:
10835:
10815:
10805:
10802:
10775:
10772:
10749:
10743:
10737:
10731:
10728:
10722:
10719:
10716:
10706:
10690:
10681:
10677:
10653:
10633:
10610:
10607:
10587:
10584:
10581:
10561:
10534:
10514:
10511:
10467:
10447:
10420:
10400:
10397:
10353:
10345:
10343:is Hausdorff.
10306:
10286:
10280:
10272:
10261:
10216:
10159:
10156:
10140:
10132:
10121:
10117:
10112:
10108:
10085:
10065:
10045:
10037:
10029:
10018:
10014:
10011:
10006:
10002:
9999:
9996:
9993:
9990:
9986:
9981:
9976:
9972:
9965:
9962:
9957:
9952:
9945:
9937:
9926:
9921:
9910:
9891:
9887:
9884:
9864:
9807:
9806:
9800:
9787:
9775:
9772:
9756:
9753:
9729:
9725:
9719:
9706:
9703:
9700:
9694:
9682:
9678:
9672:
9666:
9654:
9646:
9643:
9639:
9613:
9610:
9606:
9580:
9577:
9573:
9552:
9544:
9511:
9498:
9490:
9487:
9483:
9479:
9470:
9466:
9463:
9443:
9435:
9432:
9428:
9419:
9403:
9400:
9380:
9353:
9333:
9330:
9327:
9324:
9321:
9277:
9254:
9251:
9231:
9204:
9184:
9177:
9160:
9157:
9140:is a measure.
9127:
9107:
9098:
9092:
9066:
9060:
9054:
9044:
9041:
9021:
8986:
8983:
8973:
8970:
8964:
8937:
8934:
8914:
8870:
8862:
8844:
8832:
8829:
8826:
8820:
8814:
8811:
8799:
8796:
8777:
8771:
8768:
8759:
8733:
8727:
8721:
8718:
8711:
8710:
8697:
8688:
8682:
8654:
8645:
8642:
8630:
8627:
8620:
8619:
8605:
8585:
8573:
8570:
8559:
8553:
8548:
8545:
8542:
8538:
8531:
8528:
8502:
8499:
8496:
8487:
8484:
8481:
8461:
8458:
8438:
8394:
8386:
8385:
8384:
8367:
8364:
8344:
8332:
8329:
8326:
8320:
8314:
8308:
8292:
8289:
8257:
8254:
8251:
8240:
8223:
8220:
8211:
8194:
8191:
8176:
8163:
8154:
8150:
8147:
8127:
8107:
8081:
8077:
8074:
8048:
8024:
8018:
8012:
8003:
8000:
7991:
7978:
7975:
7972:
7966:
7960:
7951:
7941:
7938:
7932:
7923:
7917:
7908:
7905:
7896:
7883:
7880:
7877:
7871:
7865:
7856:
7846:
7843:
7837:
7827:
7824:
7805:
7794:
7791:
7782:
7774:
7760:
7745:
7742:
7726:
7723:
7704:
7700:
7697:
7694:
7688:
7685:
7682:
7676:
7664:
7655:
7647:
7633:
7621:
7618:
7602:
7599:
7580:
7576:
7570:
7557:
7554:
7551:
7545:
7539:
7527:
7518:
7510:
7509:
7508:
7488:
7467:
7459:
7453:
7440:
7431:
7422:
7406:
7403:
7359:
7332:
7316:
7302:
7299:
7256:
7240:
7224:
7219:
7216:
7206:
7203:
7183:
7180:
7172:
7132:
7129:
7109:
7101:
7072:
7056:
7040:
7035:
7032:
7022:
7019:
6999:
6996:
6988:
6948:
6945:
6925:
6917:
6898:
6895:
6872:
6842:
6834:
6819:
6811:
6810:
6793:
6760:
6730:
6707:
6701:
6698:
6695:
6692:
6684:
6670:
6667:
6664:
6661:
6638:
6635:
6629:
6623:
6603:
6578:
6575:
6556:
6535:
6532:
6529:
6506:
6503:
6497:
6491:
6471:
6464:We have that
6463:
6462:
6449:
6446:
6443:
6440:
6417:
6411:
6406:
6403:
6400:
6396:
6392:
6386:
6380:
6360:
6351:
6348:
6345:
6336:
6333:
6330:
6310:
6304:
6291:
6276:
6273:
6272:
6266:
6262:
6249:
6243:
6235:
6232:
6213:
6209:
6203:
6174:
6165:
6153:
6149:
6146:
6138:
6122:
6102:
6075:
6055:
6052:
6008:
5993:
5983:
5981:
5977:
5973:
5957:
5954:
5947:
5928:
5925:
5922:
5919:
5916:
5906:
5902:
5898:
5893:
5891:
5875:
5867:
5851:
5831:
5828:
5819:
5813:
5809:
5787:
5779:
5740:
5734:
5711:
5708:
5702:
5699:
5686:
5676:
5662:
5654:
5638:
5634:
5629:
5625:
5621:
5617:
5612:
5609:
5606:
5602:
5598:
5594:
5588:
5584:
5578:
5575:
5572:
5568:
5563:
5559:
5537:
5532:
5528:
5522:
5519:
5516:
5512:
5491:
5488:
5485:
5482:
5477:
5473:
5452:
5449:
5446:
5426:
5386:
5377:
5361:
5357:
5336:
5332:
5328:
5325:
5322:
5319:
5313:
5305:
5297:
5292:
5288:
5282:
5279:
5276:
5272:
5267:
5260:
5255:
5251:
5245:
5242:
5239:
5235:
5214:
5211:
5208:
5205:
5200:
5196:
5175:
5160:
5146:
5126:
5123:
5103:
5080:
5077:
5071:
5065:
5062:
5059:
5056:
5053:
5047:
5044:
5038:
5032:
5011:
5006:
5002:
4998:
4994:
4989:
4981:
4977:
4951:
4943:
4939:
4935:
4929:
4923:
4920:
4917:
4914:
4911:
4905:
4900:
4892:
4888:
4879:
4873:
4870:
4864:
4858:
4855:
4852:
4849:
4846:
4840:
4820:
4812:
4808:
4804:
4798:
4792:
4789:
4786:
4783:
4780:
4772:
4768:
4764:
4758:
4755:
4749:
4743:
4740:
4737:
4734:
4731:
4708:
4663:
4659:
4655:
4649:
4643:
4640:
4637:
4634:
4631:
4608:
4605:
4583:
4579:
4558:
4553:
4549:
4545:
4542:
4533:
4520:
4516:
4511:
4507:
4503:
4499:
4496:
4492:
4487:
4483:
4479:
4475:
4452:
4449:
4445:
4440:
4436:
4432:
4428:
4408:
4405:
4400:
4396:
4375:
4370:
4366:
4362:
4359:
4336:
4333:
4327:
4324:
4318:
4312:
4309:
4306:
4303:
4300:
4294:
4274:
4271:
4266:
4262:
4241:
4221:
4195:
4192:
4189:
4183:
4174:
4168:
4163:
4159:
4138:
4115:
4112:
4106:
4103:
4097:
4091:
4088:
4085:
4082:
4079:
4073:
4053:
4047:
4041:
4038:
4032:
4026:
4006:
4000:
3997:
3991:
3988:
3982:
3976:
3973:
3970:
3967:
3964:
3958:
3950:
3931:
3928:
3922:
3919:
3913:
3907:
3904:
3901:
3898:
3895:
3889:
3882:is such that
3869:
3860:
3859:as desired.
3846:
3840:
3834:
3831:
3825:
3819:
3799:
3793:
3790:
3784:
3778:
3775:
3772:
3769:
3766:
3760:
3757:
3754:
3748:
3745:
3739:
3733:
3730:
3727:
3724:
3721:
3698:
3695:
3692:
3684:
3668:
3665:
3642:
3639:
3633:
3627:
3624:
3621:
3618:
3615:
3609:
3606:
3600:
3594:
3571:
3568:
3562:
3556:
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3550:
3547:
3544:
3538:
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3529:
3523:
3510:
3508:
3492:
3483:
3477:
3471:
3468:
3461:
3442:
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3433:
3427:
3424:
3421:
3418:
3415:
3409:
3406:
3400:
3397:
3391:
3385:
3382:
3379:
3376:
3373:
3367:
3335:
3332:
3329:
3313:
3281:
3278:
3275:
3266:
3263:
3260:
3246:
3233:
3227:
3221:
3198:
3192:
3172:
3152:
3144:
3128:
3125:
3105:
3096:
3094:
3090:
3074:
3071:
3065:
3059:
3051:
3050:
3033:
3023:
3008:
2994:
2986:
2977:
2974:
2968:
2963:
2959:
2938:
2930:
2927:
2905:
2901:
2891:
2878:
2870:
2866:
2859:
2854:
2851:
2848:
2840:
2832:
2828:
2821:
2810:
2802:
2798:
2792:
2788:
2777:
2774:
2771:
2767:
2762:
2758:
2736:
2732:
2709:
2705:
2696:
2680:
2677:
2672:
2668:
2664:
2659:
2655:
2651:
2646:
2642:
2621:
2618:
2613:
2609:
2605:
2600:
2596:
2592:
2587:
2583:
2568:
2555:
2547:
2543:
2536:
2531:
2528:
2525:
2517:
2509:
2505:
2498:
2487:
2476:
2469:
2463:
2459:
2448:
2445:
2442:
2438:
2433:
2429:
2407:
2403:
2394:
2378:
2375:
2370:
2366:
2362:
2357:
2353:
2349:
2344:
2340:
2319:
2316:
2311:
2307:
2303:
2298:
2294:
2290:
2285:
2281:
2266:
2253:
2245:
2241:
2234:
2224:
2221:
2218:
2214:
2210:
2206:
2200:
2196:
2185:
2182:
2179:
2175:
2170:
2166:
2147:
2122:
2118:
2097:
2094:
2089:
2085:
2081:
2076:
2072:
2068:
2063:
2059:
2051:
2048:
2033:
2020:
2012:
2008:
2001:
1998:
1990:
1986:
1979:
1957:
1953:
1949:
1944:
1940:
1917:
1913:
1890:
1886:
1871:
1857:
1843:
1837:
1833:
1832:Gibbs measure
1830:
1827:
1824:
1823:
1822:
1820:
1816:
1812:
1808:
1804:
1799:
1797:
1793:
1792:Young measure
1789:
1788:Radon measure
1785:
1784:Baire measure
1781:
1777:
1773:
1769:
1768:Borel measure
1749:
1729:
1726:
1718:
1713:
1709:
1704:
1700:
1696:
1691:
1687:
1683:
1678:
1674:
1671:
1670:Dirac measure
1667:
1664:
1660:
1656:
1652:
1651:unit interval
1648:
1644:
1641:
1637:
1622:
1617:
1613:
1592:
1587:
1583:
1572:
1540:
1536:
1532:
1529:
1526:
1521:
1517:
1494:
1490:
1466:
1463:
1460:
1450:
1446:
1443:
1440:
1436:
1432:
1429:
1425:
1421:
1417:
1413:
1398:
1395:
1386:
1383:
1380:
1371:
1341:
1337:
1334:measure on a
1333:
1330:
1304:
1300:
1286:
1283:
1260:
1254:
1246:
1242:
1241:
1240:
1236:
1226:
1224:
1220:
1216:
1212:
1208:
1204:
1200:
1196:
1192:
1187:
1185:
1169:
1166:
1160:
1154:
1146:
1142:
1141:
1140:measure space
1121:
1118:
1112:
1109:
1099:
1094:
1092:
1068:
1067:
1044:
1041:
1029:
1027:
1026:
1009:
989:
983:
963:
954:
941:
938:
926:
906:
894:
891:
885:
879:
876:
867:
864:
858:
855:
849:
843:
823:
820:
808:
788:
765:
757:
753:
746:
736:
733:
730:
726:
722:
718:
712:
708:
697:
694:
691:
687:
682:
678:
670:
669:disjoint sets
647:
644:
641:
631:
627:
615:
611:
597:
588:
585:
571:
568:
556:
549:
535:
532:
526:
520:
511:
505:
502:
494:
491:
490:
489:
487:
483:
447:
440:
424:
421:
413:
399:
371:be a set and
358:
334:
325:
316:
314:
310:
306:
302:
301:Nikolai Luzin
298:
294:
290:
286:
282:
277:
275:
271:
267:
263:
259:
255:
251:
247:
243:
239:
235:
231:
227:
223:
219:
215:
207:
191:
188:
168:
148:
145:
137:
121:
113:
108:
98:
95:
87:
77:
73:
67:
66:
60:
55:
46:
45:
40:
33:
19:
14627:
14579:
14448:
14195:Inequalities
14135:Uniform norm
14023:
13992:
13950:
13943:
13653:Main results
13389:Set function
13317:Metric outer
13272:Decomposable
13208:
13129:Cylinder set
13049:
13042:
12964:
12944:Folland 1999
12939:
12911:
12904:
12873:
12867:
12860:Fremlin 2016
12855:
12843:
12836:Fremlin 2016
12831:
12824:Fremlin 2016
12819:
12812:Fremlin 2016
12797:Fremlin 2016
12792:
12780:
12775:, Theorem 1.
12758:Fremlin 2016
12753:
12741:
12729:
12722:Fremlin 2016
12717:
12710:Nielsen 1997
12690:
12678:
12671:Folland 1999
12666:
12640:
12634:
12613:
12603:
12596:
12584:
12552:
12533:
12530:Tao, Terence
12520:
12496:
12486:
12483:Royden, H.L.
12465:
12446:
12438:
12409:
12402:
12393:
12365:
12361:
12346:
12322:
12306:. Springer.
12303:
12293:
12271:
12252:. Springer.
12249:
12230:
12222:Chapter III.
12204:
12186:
12177:. MacMillan.
12174:
12156:
12138:
12135:Bauer, Heinz
12127:
12117:Bibliography
11751:
11591:
11587:
11583:
11575:
11573:
11519:
11515:
11507:
11505:
11492:
11478:
11477:is called a
11468:
11453:
11452:is called a
11443:
11440:set function
11437:
11405:
11387:
10872:
10869:
9515:
9513:
9190:
9182:
9149:
8927:-algebra on
8451:-algebra on
8213:
8184:
8182:
8179:Non-examples
8039:
7457:
7455:
7318:
7291:
6263:
6136:
6000:
5972:real numbers
5897:real numbers
5894:
5889:
5865:
5688:
5378:
5167:
5025:then equals
4534:
3949:monotonicity
3861:
3515:
3252:
3097:
3092:
3088:
3047:
3046:is called a
3025:
3016:Completeness
2892:
2697:of the sets
2695:intersection
2574:
2395:of the sets
2272:
2044:
1877:
1874:Monotonicity
1849:
1841:
1800:
1796:Loeb measure
1765:
1711:
1707:
1702:
1698:
1694:
1689:
1685:
1676:
1672:
1659:distribution
1658:
1654:
1435:Haar measure
1335:
1238:
1188:
1138:
1137:is called a
1095:
1090:
1064:
1063:is called a
1030:
1023:
1022:is called a
955:
780:
667:of pairwise
616:collections
586:
492:
485:
484:is called a
439:set function
350:
305:Johann Radon
278:
217:
211:
90:
84:January 2021
81:
62:
14455:Integration
14393:Von Neumann
14207:Chebyshev's
13613:compact set
13580:of measures
13516:Pushforward
13509:Projections
13499:Logarithmic
13342:Probability
13332:Pre-measure
13114:Borel space
13032:of measures
12773:Luther 1967
12588:Archimedes
12368:: 953â959.
11736:Volume form
9514:We say the
7456:We say the
7173:induced by
6989:induced by
6373:and assume
5675:-complete.
2693:) then the
2391:) then the
1706:(a), where
1555:looks like
1089:are called
612:): For all
610:-additivity
293:Ămile Borel
246:probability
214:mathematics
76:introducing
14643:Categories
14480:stochastic
14388:C*-algebra
14212:Clarkson's
13585:in measure
13312:Maximising
13282:Equivalent
13176:Vitali set
12695:Edgar 1998
12683:Edgar 1998
12625:2111.09266
12576:References
11420:Vitali set
9314:such that
6137:semifinite
5864:is called
5844:A measure
5379:A measure
5164:Additivity
4214:such that
3460:almost all
1661:. See the
1364:such that
495:: For all
319:Definition
285:Archimedes
260:, as with
59:references
14592:Functions
14383:*-algebra
14368:Quasinorm
14237:Minkowski
14128:Essential
14091:∞
13920:Lp spaces
13699:Maharam's
13669:Dominated
13482:Intensity
13477:Hausdorff
13384:Saturated
13302:Invariant
13207:Types of
13166:Ď-algebra
13136:đ-system
13102:Borel set
13097:Baire set
12971:EMS Press
12961:"Measure"
12449:. Wiley.
12384:124262782
12097:∞
12094:−
12071:μ
11996:∞
11981:∈
11969:μ
11957:⊆
11951:∀
11939:∞
11924:μ
11920:∈
11914:∃
11891:μ
11865:∞
11847:μ
11829:⊆
11823:∃
11811:∞
11796:μ
11792:∈
11786:∀
11763:μ
11540:∞
11516:countable
11359:∗
11351:μ
11308:μ
11300:∞
11248:μ
11220:μ
11197:∈
11185:μ
11173:∫
11151:↦
11140:∗
11128:μ
11099:→
11093:μ
11085:∞
10948:μ
10849:ν
10840:×
10836:μ
10806:∈
10776:∈
10744:ν
10732:μ
10720:×
10691:ν
10682:×
10678:μ
10654:ν
10634:μ
10614:∞
10611:−
10588:ν
10582:μ
10535:ν
10421:μ
10307:μ
10281:μ
10141:μ
10066:μ
10038:μ
10015:∈
10003:μ
9991:∫
9969:↦
9958:∗
9946:μ
9917:→
9911:μ
9903:∞
9776:∈
9754:≥
9730:μ
9726:∩
9707:∈
9683:μ
9679:−
9667:μ
9650:∞
9647:−
9640:μ
9617:∞
9614:−
9607:μ
9584:∞
9581:−
9574:μ
9553:μ
9530:∞
9527:−
9494:∞
9491:−
9484:μ
9471:μ
9464:μ
9439:∞
9436:−
9429:μ
9401:ξ
9354:ν
9334:ξ
9328:ν
9322:μ
9278:ξ
9258:∞
9255:−
9205:μ
9164:∞
9161:−
9128:μ
9102:∞
9093:×
9080:∖
9067:∪
9055:×
9042:μ
8974:∈
8915:σ
8836:∞
8809:∅
8797:μ
8766:∅
8692:∞
8683:×
8674:∅
8668:∖
8655:∪
8640:∅
8628:μ
8606:μ
8574:∈
8546:∈
8539:∑
8529:μ
8506:∞
8491:→
8439:σ
8371:∞
8368:−
8336:∞
8321:∈
8309:μ
8293:∈
8287:∀
8261:∞
8227:∞
8224:−
8198:∞
8195:−
8155:μ
8148:μ
8128:μ
8108:μ
8082:μ
8075:μ
8049:μ
8013:×
8007:∞
7979:∈
7961:μ
7942:∈
7933:∪
7927:∞
7918:×
7912:∞
7884:∈
7866:μ
7847:∈
7838:∪
7806:μ
7795:μ
7783:μ
7746:∈
7724:≥
7705:μ
7701:∈
7686:∩
7677:μ
7656:μ
7622:∈
7600:≥
7581:μ
7577:∩
7558:∈
7540:μ
7519:μ
7489:μ
7468:μ
7432:μ
7404:μ
7333:μ
7300:μ
7237:Then the
7207:∈
7053:Then the
7023:∈
6731:μ
6702:×
6665:∈
6642:∞
6604:μ
6533:∈
6510:∞
6472:μ
6444:⊆
6404:∈
6397:∑
6381:μ
6355:∞
6340:→
6247:∅
6244:≠
6214:μ
6210:∩
6169:∞
6154:μ
6150:∈
6123:μ
6076:μ
5974:with the
5852:μ
5829:μ
5814:μ
5788:μ
5764:∞
5735:μ
5712:μ
5706:Σ
5663:κ
5630:α
5618:μ
5613:λ
5610:∈
5607:α
5603:∑
5589:α
5579:λ
5576:∈
5573:α
5569:⋃
5560:μ
5541:Σ
5538:∈
5533:α
5523:λ
5520:∈
5517:α
5513:⋃
5492:λ
5486:α
5478:α
5453:κ
5447:λ
5427:κ
5407:Σ
5387:μ
5326:⊆
5317:∞
5280:∈
5273:∑
5243:∈
5236:∑
5212:∈
5057:∈
5048:μ
4987:↑
4915:∈
4906:μ
4898:↑
4871:≥
4850:∈
4841:μ
4784:∈
4769:⋂
4756:≥
4735:∈
4709:μ
4689:Σ
4635:∈
4456:∞
4406:≥
4340:∞
4325:≥
4304:∈
4295:μ
4272:≥
4199:∞
4184:∪
4178:∞
4175:−
4169:∈
4119:∞
4083:∈
4074:μ
4004:∞
3989:≥
3968:∈
3959:μ
3935:∞
3899:∈
3890:μ
3770:∈
3746:≥
3725:∈
3640:≥
3619:∈
3610:μ
3548:∈
3539:μ
3487:∞
3481:∞
3478:−
3472:∈
3419:∈
3410:μ
3398:≥
3377:∈
3368:μ
3339:∞
3311:Σ
3285:∞
3270:→
3222:μ
3214:to equal
3193:μ
3060:μ
2987:⊆
2981:∞
2931:∈
2860:μ
2852:≥
2822:μ
2817:∞
2814:→
2783:∞
2768:⋂
2759:μ
2681:…
2678:⊇
2665:⊇
2652:⊇
2622:…
2537:μ
2529:≥
2499:μ
2494:∞
2491:→
2454:∞
2439:⋃
2430:μ
2379:…
2376:⊆
2363:⊆
2350:⊆
2320:…
2235:μ
2230:∞
2215:∑
2211:≤
2191:∞
2176:⋃
2167:μ
2145:Σ
2098:…
2047:countable
2002:μ
1999:≤
1980:μ
1950:⊆
1858:μ
1815:conserved
1530:…
1491:μ
1414:Circular
1372:μ
1340:intervals
1255:μ
1229:Instances
1155:μ
1122:μ
1116:Σ
1077:Σ
1048:Σ
1031:The pair
1010:μ
987:∞
984:±
964:μ
933:∅
927:μ
901:∅
895:μ
880:μ
871:∅
868:∪
859:μ
844:μ
815:∅
809:μ
747:μ
742:∞
727:∑
703:∞
688:⋃
679:μ
653:∞
614:countable
598:σ
563:∅
557:μ
533:≥
521:μ
509:Σ
506:∈
468:Σ
448:μ
400:σ
379:Σ
335:μ
238:magnitude
206:empty set
14617:Infinity
14470:ordinary
14450:Calculus
14232:Markov's
14227:HĂślder's
14217:Hanner's
14034:Bessel's
13972:function
13956:Lebesgue
13716:Fubini's
13706:Egorov's
13674:Monotone
13633:variable
13611:Random:
13562:Strongly
13487:Lebesgue
13472:Harmonic
13462:Gaussian
13447:Counting
13414:Spectral
13409:Singular
13399:s-finite
13394:Ď-finite
13277:Discrete
13252:Complete
13209:Measures
13183:Null set
13071:function
12931:21196971
12532:(2011).
12270:(1999).
12229:(2002).
12185:(2006),
12137:(2001),
11598:See also
11580:ba space
11555:and the
11487:for the
11426:and the
11059:and let
10940:and let
10765:for all
10527:and let
10229:and let
9877:and let
9420:measure
9270:measure
9179:â
9034:and let
7423:element
7421:greatest
7196:and let
7012:and let
6654:for all
6522:for all
6433:for all
6068:and let
5946:integers
5944:for all
5866:Ď-finite
5227:define:
4131:for all
4019:so that
3951:implies
3812:so that
3093:complete
3049:null set
2050:sequence
2045:For any
1420:rotation
1329:complete
1219:Bourbaki
1195:analysis
412:-algebra
112:monotone
14475:partial
14252:Results
13951:Measure
13628:process
13623:measure
13618:element
13557:Bochner
13531:Trivial
13526:Tangent
13504:Product
13362:Regular
13340:)
13327:Perfect
13300:)
13265:)
13257:Content
13247:Complex
13188:Support
13161:-system
13050:Measure
12973:, 2001
12896:2840012
12126:(1995)
11588:bounded
11584:bounded
11512:content
11406:If the
10828:(Here,
8239:measure
8210:measure
7169:be the
6985:be the
6685:Taking
6277:Assume
6115:We say
1715:is the
486:measure
480:to the
283:, when
218:measure
72:improve
14612:Series
14105:spaces
14026:spaces
13995:spaces
13946:spaces
13934:Banach
13694:Jordan
13679:Vitali
13638:vector
13567:Weakly
13429:Vector
13404:Signed
13357:Random
13298:Quasi-
13287:Finite
13267:Convex
13227:Banach
13217:Atomic
13045:spaces
13030:
12929:
12919:
12894:
12884:
12563:
12540:
12503:
12472:
12453:
12425:
12382:
12333:
12310:
12278:
12256:
12237:
12215:
12193:
12163:
12145:
11576:charge
11520:finite
9185:(1985)
8183:Every
8040:Since
5139:Since
2480:
2474:
1794:, and
1645:Every
1605:where
1437:for a
1422:, and
1201:) are
1098:triple
518:
515:
311:, and
244:, and
234:volume
226:length
136:subset
61:, but
14607:Limit
13536:Young
13457:Euler
13452:Dirac
13424:Tight
13352:Radon
13322:Outer
13292:Inner
13242:Brown
13237:Borel
13232:Besov
13222:Baire
12620:arXiv
12380:S2CID
11743:Notes
11323:"is"
11240:Then
10299:Then
10058:Then
9418:least
8907:be a
8431:be a
8100:then
5653:ideal
4468:then
4352:when
3947:then
3711:then
3516:Both
3513:Proof
2393:union
1972:then
1817:(see
1680:(cf.
1416:angle
1327:is a
1213:with
1002:then
671:in ÎŁ,
460:from
414:over
134:is a
14290:For
14144:Maps
13800:For
13689:Hahn
13545:Maps
13467:Haar
13338:Sub-
13092:Atom
13080:Sets
12927:OCLC
12917:ISBN
12882:ISBN
12561:ISBN
12538:ISBN
12501:ISBN
12470:ISBN
12451:ISBN
12423:ISBN
12347:The
12331:ISBN
12308:ISBN
12276:ISBN
12254:ISBN
12235:ISBN
12213:ISBN
12191:ISBN
12161:ISBN
12143:ISBN
11859:<
11844:<
11414:are
10990:Let
10873:Let
10795:and
10646:and
10460:let
10413:let
10160:Let
9808:Let
9543:part
8950:let
8863:Let
8474:let
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8001:<
7825:>
7217:>
7145:let
7102:Let
7033:>
6961:let
6918:Let
6636:<
6576:>
6548:and
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6323:let
6233:>
6001:Let
5489:<
5450:<
5314:<
5078:>
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4805:>
4656:>
4571:let
4546:>
4535:For
4421:and
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4104:>
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3791:>
3696:<
3587:and
3569:>
3458:for
3440:>
3165:and
2951:let
1905:and
1850:Let
1803:mass
1697:) =
1668:The
1638:The
1433:The
1301:The
1243:The
1143:. A
589:(or
351:Let
268:and
242:mass
230:area
12415:doi
12370:doi
11928:pre
11800:pre
11594:.)
11034:or
11012:be
10844:cld
10686:cld
10574:If
10204:or
10182:be
9852:or
9830:be
9739:pre
9661:sup
9545:of
9366:on
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9217:on
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7860:sup
7810:pre
7709:pre
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7585:pre
7534:sup
7460:of
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6809:is
6561:pre
6218:pre
6158:pre
6135:is
5868:if
5419:is
5399:on
5264:sup
5096:if
4973:lim
4884:lim
3862:If
3300:is
3253:If
3145:of
3052:if
2845:inf
2807:lim
2575:If
2522:sup
2484:lim
2273:If
2137:in
1878:If
1719:of
1657:or
1570:det
1342:in
1305:on
1209:of
212:In
138:of
14645::
14453::
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12963:,
12925:.
12892:MR
12890:,
12880:,
12804:^
12765:^
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12649:^
12559:.
12555:.
12518:,
12421:.
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12364:.
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10157:.)
9734:sf
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3607::=
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9781:A
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9723:)
9720:A
9717:(
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9701::
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9670:(
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9467:=
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9099:+
9096:{
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9085:C
9075:A
9070:(
9064:}
9061:0
9058:{
9050:C
9045:=
9022:,
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8987:A
8984::
8979:A
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8968:{
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8842:}
8839:)
8833:+
8830:,
8827:X
8824:(
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8775:}
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8763:{
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8728:0
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8318:)
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8312:(
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8264:}
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8078:=
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7989:(
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7973::
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7967:B
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7939:A
7936:{
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7761:.
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2073:E
2069:,
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2060:E
2021:.
2018:)
2013:2
2009:E
2005:(
1996:)
1991:1
1987:E
1983:(
1958:2
1954:E
1945:1
1941:E
1918:2
1914:E
1891:1
1887:E
1838:.
1750:a
1730:.
1727:S
1712:S
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1390:]
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