2366:
2237:
1475:
or the preimage of any open set being measurable. Continuous functions, monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functions of bounded variation are all
Lebesgue measurable. A function
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Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to prove the existence of non-measurable functions. Such proofs rely on the
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2361:{\displaystyle \mathbf {1} _{A}:(X,\Sigma )\to \mathbb {R} ,\quad \mathbf {1} _{A}(x)={\begin{cases}1&{\text{ if }}x\in A\\0&{\text{ otherwise}},\end{cases}}}
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is non-metrizable. The corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence.
3618:
976:. Continuous functions are Borel functions but not all Borel functions are continuous. However, a measurable function is nearly a continuous function; see
4186:
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614:
4203:
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422:
1520:
The sum and product of two complex-valued measurable functions are measurable. So is the quotient, so long as there is no division by zero.
1387:
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3021:
2880:
1666:
1285:
2011:
Indeed, two
Lebesgue-measurable functions may be constructed in such a way as to make their composition non-Lebesgue-measurable.
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1971:
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3708:
4117:
2796:
2771:
2746:
2721:
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825:
3665:
3655:
2609: â Function spaces generalizing finite-dimensional p norm spaces - Vector spaces of measurable functions: the
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913:
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17:
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1887:
764:-algebras in the definition above is sometimes implicit and left up to the context. For example, for
4102:
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2312:
2030:
of a sequence (viz., countably many) of real-valued measurable functions are all measurable as well.
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2866:
2176:
2140:
851:
102:
1106:{\displaystyle f:(\mathbb {R} ,{\mathcal {L}})\to (\mathbb {C} ,{\mathcal {B}}_{\mathbb {C} }),}
549:
4107:
4036:
3929:
3909:
3738:
3638:
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3182:
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1455:
582:
275:
3934:
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3252:
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833:
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375:
230:
209:
53:
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3998:
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883:
76:
8:
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1030:
57:
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3488:
3187:
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2231:
2171:
2099:
2079:
1265:
977:
829:
355:
335:
188:
168:
80:
845:
Random variables are by definition measurable functions defined on probability spaces.
4026:
3660:
3396:
3357:
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3259:
3177:
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2935:
2817:
2792:
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84:
61:
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is a metric space (endowed with the Borel algebra). This is not true in general if
45:
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3406:
3401:
3379:
3237:
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3122:
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88:
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since the preimage of any point in the range is some proper, nonempty subset of
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2600:
2394:. This is a non-measurable function since the preimage of the measurable set
2391:
1191:
814:
72:
2133:
without the axiom of choice does not prove the existence of such functions.
4010:
3809:
3342:
3264:
3004:
3041:
206:
3207:
1510:
is measurable if and only if the real and imaginary parts are measurable.
817:(generated by all the open sets) is a common choice. Some authors define
33:
4263:
3954:
3051:
56:
set is measurable. This is in direct analogy to the definition that a
4258:
4243:
3033:
2977:
2972:
2034:
1445:{\displaystyle \{f\geq \alpha \},\{f<\alpha \},\{f\leq \alpha \}}
994:
3898:
3867:
3818:
3795:
3058:
2917:
2610:
2606:
2015:
821:
as exclusively real-valued ones with respect to the Borel algebra.
68:
64:
49:
2858:
2019:
27:
Function for which the preimage of a measurable set is measurable
665:{\displaystyle f\colon (X,\Sigma )\rightarrow (Y,\mathrm {T} ).}
499:{\displaystyle f^{-1}(E):=\{x\in X\mid f(x)\in E\}\in \Sigma .}
828:, other non-equivalent definitions of measurability, such as
3764:
1735:{\displaystyle g\circ f:(X,\Sigma _{1})\to (Z,\Sigma _{3}).}
2354:
1346:{\displaystyle \{f>\alpha \}=\{x\in X:f(x)>\alpha \}}
839:
75:, measurable functions are used in the definition of the
980:. If a Borel function happens to be a section of a map
2739:
Real
Analysis: Modern Techniques and their Applications
1663:
are measurable functions, then so is their composition
3960:
2616:
2570:
2547:
2506:
2486:
2452:
2426:
2400:
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2240:
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2179:
2143:
2102:
2082:
2043:
1974:
1928:
1890:
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1514:
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1359:
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1039:
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514:
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338:
310:
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253:
233:
212:
191:
171:
137:
105:
1877:{\displaystyle g:(Y,\Sigma _{3})\to (Z,\Sigma _{4})}
1807:{\displaystyle f:(X,\Sigma _{1})\to (Y,\Sigma _{2})}
1656:{\displaystyle g:(Y,\Sigma _{2})\to (Z,\Sigma _{3})}
1586:{\displaystyle f:(X,\Sigma _{1})\to (Y,\Sigma _{2})}
2814:
2812:Aliprantis, Charalambos D.; Border, Kim C. (2006).
3973:
2709:
2680:
2629:
2579:
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2472:
2435:
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2003:
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684:
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603:
567:
538:
498:
412:
384:
364:
344:
324:
296:
264:
239:
218:
197:
177:
157:
123:
2004:{\displaystyle \Sigma _{3}\subseteq \Sigma _{2}.}
1222:Lebesgue measurable functions are of interest in
44:is a function between the underlying sets of two
4283:
2811:
48:that preserves the structure of the spaces: the
67:the topological structure: the preimage of any
2480:is non-measurable with respect to the trivial
2446:As another example, any non-constant function
3780:
2874:
1183:{\displaystyle {\mathcal {B}}_{\mathbb {C} }}
3619:RieszâMarkovâKakutani representation theorem
2674:
2672:
2670:
2668:
2525:
2513:
2407:
2401:
2037:limit of a sequence of measurable functions
1884:are measurable functions, their composition
1439:
1427:
1421:
1409:
1403:
1391:
1340:
1307:
1301:
1289:
1226:because they can be integrated. In the case
539:{\displaystyle \sigma (f)\subseteq \Sigma ,}
484:
451:
2534:{\displaystyle \Sigma =\{\varnothing ,X\},}
2120:
4204:Vitale's random BrunnâMinkowski inequality
3787:
3773:
3714:Vitale's random BrunnâMinkowski inequality
2881:
2867:
2791:(2 ed.). Cambridge University Press.
2678:
2644: â Subject of study in ergodic theory
1157:-algebra of Lebesgue measurable sets, and
2805:
2707:
2665:
2466:
2376:
2275:
1961:{\displaystyle (\Sigma _{1},\Sigma _{4})}
1496:
1367:
1246:
1205:
1174:
1091:
1074:
1050:
1015:{\displaystyle Y\xrightarrow {~\pi ~} X,}
796:
772:
2701:
1377:{\displaystyle \alpha \in \mathbb {R} .}
824:If the values of the function lie in an
739:
2736:
2564:which is not an element of the trivial
2129:in an essential way, in the sense that
840:Notable classes of measurable functions
14:
4284:
2786:
2761:
2730:
1282:is Lebesgue measurable if and only if
965:{\displaystyle f:(X,\Sigma )\to (Y,T)}
304:is said to be measurable if for every
3768:
2862:
2780:
2755:
611:is a measurable function, one writes
4217:Applications & related
3727:Applications & related
1256:{\displaystyle f:X\to \mathbb {R} ,}
94:
4136:Marcinkiewicz interpolation theorem
2642:Measure-preserving dynamical system
2473:{\displaystyle f:X\to \mathbb {R} }
2230:one can construct a non-measurable
1503:{\displaystyle f:X\to \mathbb {C} }
672:to emphasize the dependency on the
165:be measurable spaces, meaning that
24:
4062:Symmetric decreasing rearrangement
3966:
2888:
2571:
2507:
2265:
2214:
2153:
1989:
1976:
1946:
1933:
1862:
1837:
1792:
1767:
1717:
1692:
1641:
1616:
1571:
1546:
1515:Properties of measurable functions
1384:This is also equivalent to any of
1167:
1122:
1084:
1059:
1033:function is a measurable function
938:
864:
720:
699:
652:
633:
530:
490:
406:
379:
318:
255:
234:
205:are sets equipped with respective
148:
115:
25:
4308:
2834:
2603: â Type of topological space
2516:
826:infinite-dimensional vector space
813:or other topological spaces, the
413:{\displaystyle E\in \mathrm {T} }
325:{\displaystyle E\in \mathrm {T} }
3656:Lebesgue differentiation theorem
3537:Carathéodory's extension theorem
2285:
2243:
2223:{\displaystyle A\notin \Sigma ,}
158:{\displaystyle (Y,\mathrm {T} )}
2282:
1915:{\displaystyle g\circ f:X\to Z}
2716:. Cambridge University Press.
2462:
2301:
2295:
2271:
2268:
2256:
2156:
2144:
2060:
1955:
1929:
1906:
1871:
1852:
1849:
1846:
1827:
1801:
1782:
1779:
1776:
1757:
1726:
1707:
1704:
1701:
1682:
1650:
1631:
1628:
1625:
1606:
1580:
1561:
1558:
1555:
1536:
1492:
1331:
1325:
1242:
1130:{\displaystyle {\mathcal {L}}}
1097:
1070:
1067:
1064:
1046:
959:
947:
944:
941:
929:
899:
887:
867:
855:
656:
642:
639:
636:
624:
595:
562:
556:
524:
518:
475:
469:
445:
439:
288:
152:
138:
118:
106:
13:
1:
4032:Convergence almost everywhere
3794:
2789:Real Analysis and Probability
1215:{\displaystyle \mathbb {C} .}
806:{\displaystyle \mathbb {C} ,}
782:{\displaystyle \mathbb {R} ,}
730:{\displaystyle \mathrm {T} .}
265:{\displaystyle \mathrm {T} .}
83:, a measurable function on a
2383:{\displaystyle \mathbb {R} }
2069:{\displaystyle f_{n}:X\to Y}
7:
4199:PrĂ©kopaâLeindler inequality
4052:Locally integrable function
3974:{\displaystyle L^{\infty }}
3709:PrĂ©kopaâLeindler inequality
2854:Encyclopedia of Mathematics
2845:Encyclopedia of Mathematics
2737:Folland, Gerald B. (1999).
2679:Strichartz, Robert (2000).
2596:Bochner measurable function
2589:
2390:is equipped with the usual
2195:{\displaystyle A\subset X,}
2162:{\displaystyle (X,\Sigma )}
2131:ZermeloâFraenkel set theory
873:{\displaystyle (X,\Sigma )}
124:{\displaystyle (X,\Sigma )}
10:
4313:
3945:Square-integrable function
3651:Lebesgue's density theorem
2653:Weakly measurable function
568:{\displaystyle \sigma (f)}
4216:
4194:MinkowskiâSteiner formula
4164:
4126:
4070:
4019:
3953:
3897:
3866:
3802:
3726:
3704:MinkowskiâSteiner formula
3674:
3634:
3627:
3527:
3519:Projection-valued measure
3420:
3313:
3082:
2955:
2896:
2708:Carothers, N. L. (2000).
1452:being measurable for all
4177:Isoperimetric inequality
3687:Isoperimetric inequality
3666:VitaliâHahnâSaks theorem
2995:Carathéodory's criterion
2816:(3 ed.). Springer.
2658:
2580:{\displaystyle \Sigma .}
2121:Non-measurable functions
1468:{\displaystyle \alpha ,}
916:, a measurable function
604:{\displaystyle f:X\to Y}
577:Ï-algebra generated by f
297:{\displaystyle f:X\to Y}
4182:BrunnâMinkowski theorem
3692:BrunnâMinkowski theorem
3561:Decomposition theorems
2493:{\displaystyle \sigma }
1150:{\displaystyle \sigma }
757:{\displaystyle \sigma }
705:{\displaystyle \Sigma }
685:{\displaystyle \sigma }
385:{\displaystyle \Sigma }
240:{\displaystyle \Sigma }
219:{\displaystyle \sigma }
4037:Convergence in measure
3975:
3739:Descriptive set theory
3639:Disintegration theorem
3074:Universally measurable
2787:Dudley, R. M. (2002).
2762:Royden, H. L. (1988).
2687:. Jones and Bartlett.
2631:
2581:
2558:
2535:
2494:
2474:
2437:
2420:is the non-measurable
2414:
2384:
2362:
2224:
2196:
2163:
2110:
2090:
2070:
2005:
1962:
1916:
1878:
1808:
1736:
1657:
1587:
1504:
1469:
1446:
1378:
1353:is measurable for all
1347:
1276:
1257:
1216:
1184:
1151:
1131:
1107:
1016:
966:
906:
874:
807:
783:
758:
731:
706:
686:
666:
605:
569:
540:
500:
414:
386:
366:
346:
326:
298:
266:
241:
220:
199:
179:
159:
125:
4151:RieszâFischer theorem
3976:
3935:Polarization identity
3541:Convergence theorems
3000:Cylindrical Ï-algebra
2632:
2630:{\displaystyle L^{p}}
2582:
2559:
2536:
2495:
2475:
2438:
2415:
2413:{\displaystyle \{1\}}
2385:
2363:
2225:
2197:
2164:
2136:In any measure space
2111:
2091:
2076:is measurable, where
2071:
2006:
1963:
1917:
1879:
1809:
1737:
1658:
1588:
1505:
1470:
1447:
1379:
1348:
1277:
1258:
1224:mathematical analysis
1217:
1185:
1152:
1132:
1108:
1017:
967:
907:
905:{\displaystyle (Y,T)}
875:
834:Bochner measurability
808:
784:
759:
740:Term usage variations
732:
707:
687:
667:
606:
570:
541:
501:
415:
387:
367:
347:
327:
299:
267:
242:
221:
200:
180:
160:
126:
4156:RieszâThorin theorem
3999:Infimum and supremum
3958:
3884:Lebesgue integration
3609:Minkowski inequality
3483:Cylinder set measure
3368:Infinite-dimensional
2983:equivalence relation
2913:Lebesgue integration
2614:
2568:
2545:
2504:
2484:
2450:
2424:
2398:
2372:
2238:
2205:
2177:
2141:
2100:
2080:
2041:
1972:
1926:
1888:
1818:
1748:
1667:
1597:
1527:
1480:
1456:
1388:
1357:
1286:
1266:
1230:
1201:
1161:
1141:
1117:
1037:
984:
920:
884:
852:
819:measurable functions
792:
768:
748:
716:
696:
676:
615:
583:
550:
512:
423:
396:
376:
356:
336:
308:
276:
251:
231:
210:
189:
169:
135:
103:
36:, and in particular
4118:Young's convolution
4057:Measurable function
3940:Pythagorean theorem
3930:Parseval's identity
3879:Integrable function
3604:Hölder's inequality
3466:of random variables
3428:Measurable function
3315:Particular measures
2904:Absolute continuity
2841:Measurable function
2683:The Way of Analysis
1968:-measurable unless
1031:Lebesgue measurable
1004:
392:; that is, for all
42:measurable function
4297:Types of functions
4239:Probability theory
4141:Plancherel theorem
4047:Integral transform
3994:Chebyshev distance
3971:
3920:Euclidean distance
3853:Minkowski distance
3744:Probability theory
3069:Transverse measure
3047:Non-measurable set
3029:Locally measurable
2627:
2577:
2557:{\displaystyle X,}
2554:
2531:
2490:
2470:
2436:{\displaystyle A.}
2433:
2410:
2380:
2358:
2353:
2232:indicator function
2220:
2192:
2172:non-measurable set
2159:
2106:
2086:
2066:
2001:
1958:
1912:
1874:
1804:
1732:
1653:
1583:
1500:
1465:
1442:
1374:
1343:
1272:
1253:
1212:
1180:
1147:
1127:
1103:
1012:
962:
902:
870:
830:weak measurability
803:
779:
754:
727:
702:
682:
662:
601:
565:
536:
496:
410:
382:
362:
342:
322:
294:
262:
237:
216:
195:
175:
155:
121:
81:probability theory
62:topological spaces
4279:
4278:
4212:
4211:
4027:Almost everywhere
3812: &
3762:
3761:
3722:
3721:
3451:almost everywhere
3397:Spherical measure
3295:Strictly positive
3223:Projection-valued
2963:Almost everywhere
2936:Probability space
2823:978-3-540-29587-7
2766:. Prentice Hall.
2346:
2323:
2109:{\displaystyle Y}
2089:{\displaystyle Y}
1275:{\displaystyle f}
1005:
1003:
997:
972:is also called a
365:{\displaystyle f}
345:{\displaystyle E}
332:the pre-image of
198:{\displaystyle Y}
178:{\displaystyle X}
95:Formal definition
85:probability space
77:Lebesgue integral
60:function between
46:measurable spaces
16:(Redirected from
4304:
4229:Fourier analysis
4187:Milman's reverse
4170:
4168:Lebesgue measure
4162:
4161:
4146:RiemannâLebesgue
3989:Bounded function
3980:
3978:
3977:
3972:
3970:
3969:
3889:Taxicab geometry
3844:Measurable space
3789:
3782:
3775:
3766:
3765:
3697:Milman's reverse
3680:
3678:Lebesgue measure
3632:
3631:
3036:
3022:infimum/supremum
2943:Measurable space
2883:
2876:
2869:
2860:
2859:
2828:
2827:
2809:
2803:
2802:
2784:
2778:
2777:
2759:
2753:
2752:
2734:
2728:
2727:
2715:
2705:
2699:
2698:
2686:
2676:
2636:
2634:
2633:
2628:
2626:
2625:
2586:
2584:
2583:
2578:
2563:
2561:
2560:
2555:
2540:
2538:
2537:
2532:
2499:
2497:
2496:
2491:
2479:
2477:
2476:
2471:
2469:
2442:
2440:
2439:
2434:
2419:
2417:
2416:
2411:
2389:
2387:
2386:
2381:
2379:
2367:
2365:
2364:
2359:
2357:
2356:
2347:
2344:
2324:
2321:
2294:
2293:
2288:
2278:
2252:
2251:
2246:
2229:
2227:
2226:
2221:
2201:
2199:
2198:
2193:
2168:
2166:
2165:
2160:
2115:
2113:
2112:
2107:
2095:
2093:
2092:
2087:
2075:
2073:
2072:
2067:
2053:
2052:
2014:The (pointwise)
2010:
2008:
2007:
2002:
1997:
1996:
1984:
1983:
1967:
1965:
1964:
1959:
1954:
1953:
1941:
1940:
1921:
1919:
1918:
1913:
1883:
1881:
1880:
1875:
1870:
1869:
1845:
1844:
1813:
1811:
1810:
1805:
1800:
1799:
1775:
1774:
1741:
1739:
1738:
1733:
1725:
1724:
1700:
1699:
1662:
1660:
1659:
1654:
1649:
1648:
1624:
1623:
1592:
1590:
1589:
1584:
1579:
1578:
1554:
1553:
1509:
1507:
1506:
1501:
1499:
1474:
1472:
1471:
1466:
1451:
1449:
1448:
1443:
1383:
1381:
1380:
1375:
1370:
1352:
1350:
1349:
1344:
1281:
1279:
1278:
1273:
1262:
1260:
1259:
1254:
1249:
1221:
1219:
1218:
1213:
1208:
1189:
1187:
1186:
1181:
1179:
1178:
1177:
1171:
1170:
1156:
1154:
1153:
1148:
1136:
1134:
1133:
1128:
1126:
1125:
1112:
1110:
1109:
1104:
1096:
1095:
1094:
1088:
1087:
1077:
1063:
1062:
1053:
1021:
1019:
1018:
1013:
1001:
995:
990:
971:
969:
968:
963:
911:
909:
908:
903:
879:
877:
876:
871:
812:
810:
809:
804:
799:
788:
786:
785:
780:
775:
763:
761:
760:
755:
736:
734:
733:
728:
723:
711:
709:
708:
703:
691:
689:
688:
683:
671:
669:
668:
663:
655:
610:
608:
607:
602:
574:
572:
571:
566:
545:
543:
542:
537:
505:
503:
502:
497:
438:
437:
419:
417:
416:
411:
409:
391:
389:
388:
383:
371:
369:
368:
363:
351:
349:
348:
343:
331:
329:
328:
323:
321:
303:
301:
300:
295:
271:
269:
268:
263:
258:
246:
244:
243:
238:
225:
223:
222:
217:
204:
202:
201:
196:
184:
182:
181:
176:
164:
162:
161:
156:
151:
130:
128:
127:
122:
21:
4312:
4311:
4307:
4306:
4305:
4303:
4302:
4301:
4282:
4281:
4280:
4275:
4208:
4165:
4160:
4122:
4098:HausdorffâYoung
4078:BabenkoâBeckner
4066:
4015:
3965:
3961:
3959:
3956:
3955:
3949:
3893:
3862:
3858:Sequence spaces
3798:
3793:
3763:
3758:
3754:Spectral theory
3734:Convex analysis
3718:
3675:
3670:
3623:
3523:
3471:in distribution
3416:
3309:
3139:Logarithmically
3078:
3034:
3017:Essential range
2951:
2892:
2887:
2837:
2832:
2831:
2824:
2810:
2806:
2799:
2785:
2781:
2774:
2760:
2756:
2749:
2735:
2731:
2724:
2706:
2702:
2695:
2677:
2666:
2661:
2621:
2617:
2615:
2612:
2611:
2592:
2569:
2566:
2565:
2546:
2543:
2542:
2505:
2502:
2501:
2485:
2482:
2481:
2465:
2451:
2448:
2447:
2425:
2422:
2421:
2399:
2396:
2395:
2375:
2373:
2370:
2369:
2352:
2351:
2345: otherwise
2343:
2341:
2335:
2334:
2320:
2318:
2308:
2307:
2289:
2284:
2283:
2274:
2247:
2242:
2241:
2239:
2236:
2235:
2206:
2203:
2202:
2178:
2175:
2174:
2142:
2139:
2138:
2127:axiom of choice
2123:
2101:
2098:
2097:
2081:
2078:
2077:
2048:
2044:
2042:
2039:
2038:
1992:
1988:
1979:
1975:
1973:
1970:
1969:
1949:
1945:
1936:
1932:
1927:
1924:
1923:
1889:
1886:
1885:
1865:
1861:
1840:
1836:
1819:
1816:
1815:
1795:
1791:
1770:
1766:
1749:
1746:
1745:
1720:
1716:
1695:
1691:
1668:
1665:
1664:
1644:
1640:
1619:
1615:
1598:
1595:
1594:
1574:
1570:
1549:
1545:
1528:
1525:
1524:
1517:
1495:
1481:
1478:
1477:
1457:
1454:
1453:
1389:
1386:
1385:
1366:
1358:
1355:
1354:
1287:
1284:
1283:
1267:
1264:
1263:
1245:
1231:
1228:
1227:
1204:
1202:
1199:
1198:
1196:complex numbers
1173:
1172:
1166:
1165:
1164:
1162:
1159:
1158:
1142:
1139:
1138:
1121:
1120:
1118:
1115:
1114:
1090:
1089:
1083:
1082:
1081:
1073:
1058:
1057:
1049:
1038:
1035:
1034:
1022:it is called a
985:
982:
981:
978:Luzin's theorem
921:
918:
917:
885:
882:
881:
853:
850:
849:
842:
795:
793:
790:
789:
771:
769:
766:
765:
749:
746:
745:
742:
719:
717:
714:
713:
697:
694:
693:
677:
674:
673:
651:
616:
613:
612:
584:
581:
580:
551:
548:
547:
513:
510:
509:
430:
426:
424:
421:
420:
405:
397:
394:
393:
377:
374:
373:
357:
354:
353:
337:
334:
333:
317:
309:
306:
305:
277:
274:
273:
254:
252:
249:
248:
232:
229:
228:
211:
208:
207:
190:
187:
186:
170:
167:
166:
147:
136:
133:
132:
104:
101:
100:
97:
89:random variable
28:
23:
22:
15:
12:
11:
5:
4310:
4300:
4299:
4294:
4292:Measure theory
4277:
4276:
4274:
4273:
4272:
4271:
4266:
4256:
4251:
4246:
4241:
4236:
4231:
4226:
4220:
4218:
4214:
4213:
4210:
4209:
4207:
4206:
4201:
4196:
4191:
4190:
4189:
4179:
4173:
4171:
4159:
4158:
4153:
4148:
4143:
4138:
4132:
4130:
4124:
4123:
4121:
4120:
4115:
4110:
4105:
4100:
4095:
4090:
4085:
4080:
4074:
4072:
4068:
4067:
4065:
4064:
4059:
4054:
4049:
4044:
4042:Function space
4039:
4034:
4029:
4023:
4021:
4017:
4016:
4014:
4013:
4008:
4007:
4006:
3996:
3991:
3985:
3983:
3968:
3964:
3951:
3950:
3948:
3947:
3942:
3937:
3932:
3927:
3922:
3917:
3915:CauchyâSchwarz
3912:
3906:
3904:
3895:
3894:
3892:
3891:
3886:
3881:
3875:
3873:
3864:
3863:
3861:
3860:
3855:
3850:
3841:
3836:
3835:
3834:
3824:
3816:
3814:Hilbert spaces
3806:
3804:
3803:Basic concepts
3800:
3799:
3792:
3791:
3784:
3777:
3769:
3760:
3759:
3757:
3756:
3751:
3746:
3741:
3736:
3730:
3728:
3724:
3723:
3720:
3719:
3717:
3716:
3711:
3706:
3701:
3700:
3699:
3689:
3683:
3681:
3672:
3671:
3669:
3668:
3663:
3661:Sard's theorem
3658:
3653:
3648:
3647:
3646:
3644:Lifting theory
3635:
3629:
3625:
3624:
3622:
3621:
3616:
3611:
3606:
3601:
3600:
3599:
3597:FubiniâTonelli
3589:
3584:
3579:
3578:
3577:
3572:
3567:
3559:
3558:
3557:
3552:
3547:
3539:
3533:
3531:
3525:
3524:
3522:
3521:
3516:
3511:
3506:
3501:
3496:
3491:
3485:
3480:
3479:
3478:
3476:in probability
3473:
3463:
3458:
3453:
3447:
3446:
3445:
3440:
3435:
3424:
3422:
3418:
3417:
3415:
3414:
3409:
3404:
3399:
3394:
3389:
3388:
3387:
3377:
3372:
3371:
3370:
3360:
3355:
3350:
3345:
3340:
3335:
3330:
3325:
3319:
3317:
3311:
3310:
3308:
3307:
3302:
3297:
3292:
3287:
3282:
3277:
3272:
3267:
3262:
3257:
3256:
3255:
3250:
3245:
3235:
3230:
3225:
3220:
3210:
3205:
3200:
3195:
3190:
3185:
3183:Locally finite
3180:
3170:
3165:
3160:
3155:
3150:
3145:
3135:
3130:
3125:
3120:
3115:
3110:
3105:
3100:
3095:
3089:
3087:
3080:
3079:
3077:
3076:
3071:
3066:
3061:
3056:
3055:
3054:
3044:
3039:
3031:
3026:
3025:
3024:
3014:
3009:
3008:
3007:
2997:
2992:
2987:
2986:
2985:
2975:
2970:
2965:
2959:
2957:
2953:
2952:
2950:
2949:
2940:
2939:
2938:
2928:
2923:
2915:
2910:
2900:
2898:
2897:Basic concepts
2894:
2893:
2890:Measure theory
2886:
2885:
2878:
2871:
2863:
2857:
2856:
2850:Borel function
2847:
2836:
2835:External links
2833:
2830:
2829:
2822:
2804:
2797:
2779:
2772:
2754:
2747:
2729:
2722:
2700:
2693:
2663:
2662:
2660:
2657:
2656:
2655:
2650:
2648:Vector measure
2645:
2639:
2624:
2620:
2604:
2598:
2591:
2588:
2576:
2573:
2553:
2550:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2489:
2468:
2464:
2461:
2458:
2455:
2432:
2429:
2409:
2406:
2403:
2378:
2355:
2350:
2342:
2340:
2337:
2336:
2333:
2330:
2327:
2322: if
2319:
2317:
2314:
2313:
2311:
2306:
2303:
2300:
2297:
2292:
2287:
2281:
2277:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2250:
2245:
2219:
2216:
2213:
2210:
2191:
2188:
2185:
2182:
2158:
2155:
2152:
2149:
2146:
2122:
2119:
2118:
2117:
2105:
2085:
2065:
2062:
2059:
2056:
2051:
2047:
2031:
2028:limit inferior
2024:limit superior
2012:
2000:
1995:
1991:
1987:
1982:
1978:
1957:
1952:
1948:
1944:
1939:
1935:
1931:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1873:
1868:
1864:
1860:
1857:
1854:
1851:
1848:
1843:
1839:
1835:
1832:
1829:
1826:
1823:
1803:
1798:
1794:
1790:
1787:
1784:
1781:
1778:
1773:
1769:
1765:
1762:
1759:
1756:
1753:
1742:
1731:
1728:
1723:
1719:
1715:
1712:
1709:
1706:
1703:
1698:
1694:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1652:
1647:
1643:
1639:
1636:
1633:
1630:
1627:
1622:
1618:
1614:
1611:
1608:
1605:
1602:
1582:
1577:
1573:
1569:
1566:
1563:
1560:
1557:
1552:
1548:
1544:
1541:
1538:
1535:
1532:
1521:
1516:
1513:
1512:
1511:
1498:
1494:
1491:
1488:
1485:
1464:
1461:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1373:
1369:
1365:
1362:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1271:
1252:
1248:
1244:
1241:
1238:
1235:
1211:
1207:
1176:
1169:
1146:
1124:
1102:
1099:
1093:
1086:
1080:
1076:
1072:
1069:
1066:
1061:
1056:
1052:
1048:
1045:
1042:
1027:
1011:
1008:
1000:
993:
989:
974:Borel function
961:
958:
955:
952:
949:
946:
943:
940:
937:
934:
931:
928:
925:
901:
898:
895:
892:
889:
869:
866:
863:
860:
857:
846:
841:
838:
802:
798:
778:
774:
753:
744:The choice of
741:
738:
726:
722:
701:
681:
661:
658:
654:
650:
647:
644:
641:
638:
635:
632:
629:
626:
623:
620:
600:
597:
594:
591:
588:
564:
561:
558:
555:
535:
532:
529:
526:
523:
520:
517:
495:
492:
489:
486:
483:
480:
477:
474:
471:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
436:
433:
429:
408:
404:
401:
381:
361:
341:
320:
316:
313:
293:
290:
287:
284:
281:
261:
257:
236:
215:
194:
174:
154:
150:
146:
143:
140:
120:
117:
114:
111:
108:
96:
93:
87:is known as a
38:measure theory
26:
18:Borel function
9:
6:
4:
3:
2:
4309:
4298:
4295:
4293:
4290:
4289:
4287:
4270:
4267:
4265:
4262:
4261:
4260:
4257:
4255:
4254:Sobolev space
4252:
4250:
4249:Real analysis
4247:
4245:
4242:
4240:
4237:
4235:
4234:Lorentz space
4232:
4230:
4227:
4225:
4224:Bochner space
4222:
4221:
4219:
4215:
4205:
4202:
4200:
4197:
4195:
4192:
4188:
4185:
4184:
4183:
4180:
4178:
4175:
4174:
4172:
4169:
4163:
4157:
4154:
4152:
4149:
4147:
4144:
4142:
4139:
4137:
4134:
4133:
4131:
4129:
4125:
4119:
4116:
4114:
4111:
4109:
4106:
4104:
4101:
4099:
4096:
4094:
4091:
4089:
4086:
4084:
4081:
4079:
4076:
4075:
4073:
4069:
4063:
4060:
4058:
4055:
4053:
4050:
4048:
4045:
4043:
4040:
4038:
4035:
4033:
4030:
4028:
4025:
4024:
4022:
4018:
4012:
4009:
4005:
4002:
4001:
4000:
3997:
3995:
3992:
3990:
3987:
3986:
3984:
3982:
3962:
3952:
3946:
3943:
3941:
3938:
3936:
3933:
3931:
3928:
3926:
3925:Hilbert space
3923:
3921:
3918:
3916:
3913:
3911:
3908:
3907:
3905:
3903:
3901:
3896:
3890:
3887:
3885:
3882:
3880:
3877:
3876:
3874:
3872:
3870:
3865:
3859:
3856:
3854:
3851:
3849:
3845:
3842:
3840:
3839:Measure space
3837:
3833:
3830:
3829:
3828:
3825:
3823:
3821:
3817:
3815:
3811:
3808:
3807:
3805:
3801:
3797:
3790:
3785:
3783:
3778:
3776:
3771:
3770:
3767:
3755:
3752:
3750:
3749:Real analysis
3747:
3745:
3742:
3740:
3737:
3735:
3732:
3731:
3729:
3725:
3715:
3712:
3710:
3707:
3705:
3702:
3698:
3695:
3694:
3693:
3690:
3688:
3685:
3684:
3682:
3679:
3673:
3667:
3664:
3662:
3659:
3657:
3654:
3652:
3649:
3645:
3642:
3641:
3640:
3637:
3636:
3633:
3630:
3628:Other results
3626:
3620:
3617:
3615:
3614:RadonâNikodym
3612:
3610:
3607:
3605:
3602:
3598:
3595:
3594:
3593:
3590:
3588:
3587:Fatou's lemma
3585:
3583:
3580:
3576:
3573:
3571:
3568:
3566:
3563:
3562:
3560:
3556:
3553:
3551:
3548:
3546:
3543:
3542:
3540:
3538:
3535:
3534:
3532:
3530:
3526:
3520:
3517:
3515:
3512:
3510:
3507:
3505:
3502:
3500:
3497:
3495:
3492:
3490:
3486:
3484:
3481:
3477:
3474:
3472:
3469:
3468:
3467:
3464:
3462:
3459:
3457:
3454:
3452:
3449:Convergence:
3448:
3444:
3441:
3439:
3436:
3434:
3431:
3430:
3429:
3426:
3425:
3423:
3419:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3386:
3383:
3382:
3381:
3378:
3376:
3373:
3369:
3366:
3365:
3364:
3361:
3359:
3356:
3354:
3351:
3349:
3346:
3344:
3341:
3339:
3336:
3334:
3331:
3329:
3326:
3324:
3321:
3320:
3318:
3316:
3312:
3306:
3303:
3301:
3298:
3296:
3293:
3291:
3288:
3286:
3283:
3281:
3278:
3276:
3273:
3271:
3268:
3266:
3263:
3261:
3258:
3254:
3253:Outer regular
3251:
3249:
3248:Inner regular
3246:
3244:
3243:Borel regular
3241:
3240:
3239:
3236:
3234:
3231:
3229:
3226:
3224:
3221:
3219:
3215:
3211:
3209:
3206:
3204:
3201:
3199:
3196:
3194:
3191:
3189:
3186:
3184:
3181:
3179:
3175:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3146:
3144:
3140:
3136:
3134:
3131:
3129:
3126:
3124:
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3091:
3090:
3088:
3086:
3081:
3075:
3072:
3070:
3067:
3065:
3062:
3060:
3057:
3053:
3050:
3049:
3048:
3045:
3043:
3040:
3038:
3032:
3030:
3027:
3023:
3020:
3019:
3018:
3015:
3013:
3010:
3006:
3003:
3002:
3001:
2998:
2996:
2993:
2991:
2988:
2984:
2981:
2980:
2979:
2976:
2974:
2971:
2969:
2966:
2964:
2961:
2960:
2958:
2954:
2948:
2944:
2941:
2937:
2934:
2933:
2932:
2931:Measure space
2929:
2927:
2924:
2922:
2920:
2916:
2914:
2911:
2909:
2905:
2902:
2901:
2899:
2895:
2891:
2884:
2879:
2877:
2872:
2870:
2865:
2864:
2861:
2855:
2851:
2848:
2846:
2842:
2839:
2838:
2825:
2819:
2815:
2808:
2800:
2798:0-521-00754-2
2794:
2790:
2783:
2775:
2773:0-02-404151-3
2769:
2765:
2764:Real Analysis
2758:
2750:
2748:0-471-31716-0
2744:
2740:
2733:
2725:
2723:0-521-49756-6
2719:
2714:
2713:
2712:Real Analysis
2704:
2696:
2694:0-7637-1497-6
2690:
2685:
2684:
2675:
2673:
2671:
2669:
2664:
2654:
2651:
2649:
2646:
2643:
2640:
2638:
2622:
2618:
2608:
2605:
2602:
2601:Bochner space
2599:
2597:
2594:
2593:
2587:
2574:
2551:
2548:
2528:
2522:
2519:
2510:
2487:
2459:
2456:
2453:
2444:
2430:
2427:
2404:
2393:
2392:Borel algebra
2348:
2338:
2331:
2328:
2325:
2315:
2309:
2304:
2298:
2290:
2279:
2262:
2259:
2253:
2248:
2233:
2217:
2211:
2208:
2189:
2186:
2183:
2180:
2173:
2169:
2150:
2147:
2134:
2132:
2128:
2103:
2083:
2063:
2057:
2054:
2049:
2045:
2036:
2032:
2029:
2025:
2021:
2017:
2013:
1998:
1993:
1985:
1980:
1950:
1942:
1937:
1909:
1903:
1900:
1897:
1894:
1891:
1866:
1858:
1855:
1841:
1833:
1830:
1824:
1821:
1796:
1788:
1785:
1771:
1763:
1760:
1754:
1751:
1743:
1729:
1721:
1713:
1710:
1696:
1688:
1685:
1679:
1676:
1673:
1670:
1645:
1637:
1634:
1620:
1612:
1609:
1603:
1600:
1575:
1567:
1564:
1550:
1542:
1539:
1533:
1530:
1522:
1519:
1518:
1489:
1486:
1483:
1462:
1459:
1436:
1433:
1430:
1424:
1418:
1415:
1412:
1406:
1400:
1397:
1394:
1371:
1363:
1360:
1337:
1334:
1328:
1322:
1319:
1316:
1313:
1310:
1304:
1298:
1295:
1292:
1269:
1250:
1239:
1236:
1233:
1225:
1209:
1197:
1193:
1192:Borel algebra
1144:
1100:
1078:
1054:
1043:
1040:
1032:
1028:
1025:
1024:Borel section
1009:
1006:
998:
991:
987:
979:
975:
956:
953:
950:
935:
932:
926:
923:
915:
896:
893:
890:
861:
858:
847:
844:
843:
837:
835:
831:
827:
822:
820:
816:
815:Borel algebra
800:
776:
751:
737:
724:
679:
659:
648:
645:
630:
627:
621:
618:
598:
592:
589:
586:
578:
559:
553:
533:
527:
521:
515:
506:
493:
487:
481:
478:
472:
466:
463:
460:
457:
454:
448:
442:
434:
431:
427:
402:
399:
359:
339:
314:
311:
291:
285:
282:
279:
259:
227:
213:
192:
172:
144:
141:
112:
109:
92:
90:
86:
82:
78:
74:
73:real analysis
70:
66:
63:
59:
55:
51:
47:
43:
39:
35:
30:
19:
4071:Inequalities
4056:
4011:Uniform norm
3899:
3868:
3847:
3819:
3529:Main results
3427:
3265:Set function
3193:Metric outer
3148:Decomposable
3005:Cylinder set
2946:
2918:
2813:
2807:
2788:
2782:
2763:
2757:
2738:
2732:
2711:
2703:
2682:
2445:
2137:
2135:
2124:
1922:need not be
1023:
973:
914:Borel spaces
823:
818:
743:
507:
98:
71:is open. In
41:
31:
29:
4269:Von Neumann
4083:Chebyshev's
3489:compact set
3456:of measures
3392:Pushforward
3385:Projections
3375:Logarithmic
3218:Probability
3208:Pre-measure
2990:Borel space
2908:of measures
272:A function
34:mathematics
4286:Categories
4264:C*-algebra
4088:Clarkson's
3461:in measure
3188:Maximising
3158:Equivalent
3052:Vitali set
692:-algebras
58:continuous
54:measurable
4259:*-algebra
4244:Quasinorm
4113:Minkowski
4004:Essential
3967:∞
3796:Lp spaces
3575:Maharam's
3545:Dominated
3358:Intensity
3353:Hausdorff
3260:Saturated
3178:Invariant
3083:Types of
3042:Ï-algebra
3012:đ-system
2978:Borel set
2973:Baire set
2741:. Wiley.
2572:Σ
2517:∅
2508:Σ
2500:-algebra
2488:σ
2463:→
2329:∈
2272:→
2266:Σ
2215:Σ
2212:∉
2184:⊂
2154:Σ
2061:→
2035:pointwise
1990:Σ
1986:⊆
1977:Σ
1947:Σ
1934:Σ
1907:→
1895:∘
1863:Σ
1850:→
1838:Σ
1793:Σ
1780:→
1768:Σ
1718:Σ
1705:→
1693:Σ
1674:∘
1642:Σ
1629:→
1617:Σ
1572:Σ
1559:→
1547:Σ
1493:→
1460:α
1437:α
1434:≤
1419:α
1401:α
1398:≥
1364:∈
1361:α
1338:α
1314:∈
1299:α
1243:→
1145:σ
1068:→
999:π
945:→
939:Σ
865:Σ
836:, exist.
752:σ
700:Σ
680:σ
640:→
634:Σ
622::
596:→
554:σ
531:Σ
528:⊆
516:σ
508:That is,
491:Σ
488:∈
479:∈
464:∣
458:∈
432:−
403:∈
380:Σ
315:∈
289:→
235:Σ
226:-algebras
214:σ
116:Σ
65:preserves
4108:Markov's
4103:Hölder's
4093:Hanner's
3910:Bessel's
3848:function
3832:Lebesgue
3592:Fubini's
3582:Egorov's
3550:Monotone
3509:variable
3487:Random:
3438:Strongly
3363:Lebesgue
3348:Harmonic
3338:Gaussian
3323:Counting
3290:Spectral
3285:Singular
3275:s-finite
3270:Ï-finite
3153:Discrete
3128:Complete
3085:Measures
3059:Null set
2947:function
2607:Lp space
2590:See also
2016:supremum
992:→
69:open set
50:preimage
4128:Results
3827:Measure
3504:process
3499:measure
3494:element
3433:Bochner
3407:Trivial
3402:Tangent
3380:Product
3238:Regular
3216:)
3203:Perfect
3176:)
3141:)
3133:Content
3123:Complex
3064:Support
3037:-system
2926:Measure
2443:
2170:with a
2020:infimum
1194:on the
1190:is the
1137:is the
575:is the
52:of any
3981:spaces
3902:spaces
3871:spaces
3822:spaces
3810:Banach
3570:Jordan
3555:Vitali
3514:vector
3443:Weakly
3305:Vector
3280:Signed
3233:Random
3174:Quasi-
3163:Finite
3143:Convex
3103:Banach
3093:Atomic
2921:spaces
2906:
2820:
2795:
2770:
2745:
2720:
2691:
2637:spaces
2368:where
2026:, and
1113:where
1002:
996:
546:where
372:is in
352:under
3412:Young
3333:Euler
3328:Dirac
3300:Tight
3228:Radon
3198:Outer
3168:Inner
3118:Brown
3113:Borel
3108:Besov
3098:Baire
2659:Notes
579:. If
79:. In
4166:For
4020:Maps
3676:For
3565:Hahn
3421:Maps
3343:Haar
3214:Sub-
2968:Atom
2956:Sets
2818:ISBN
2793:ISBN
2768:ISBN
2743:ISBN
2718:ISBN
2689:ISBN
2033:The
1814:and
1593:and
1416:<
1335:>
1296:>
912:are
880:and
832:and
712:and
247:and
185:and
131:and
99:Let
40:, a
2852:at
2843:at
2234::
1744:If
1523:If
848:If
32:In
4288::
2667:^
2022:,
2018:,
1029:A
449::=
91:.
3963:L
3900:L
3869:L
3846:/
3820:L
3788:e
3781:t
3774:v
3212:(
3172:(
3137:(
3035:Ï
2945:/
2919:L
2882:e
2875:t
2868:v
2826:.
2801:.
2776:.
2751:.
2726:.
2697:.
2623:p
2619:L
2575:.
2552:,
2549:X
2529:,
2526:}
2523:X
2520:,
2514:{
2511:=
2467:R
2460:X
2457::
2454:f
2431:.
2428:A
2408:}
2405:1
2402:{
2377:R
2349:,
2339:0
2332:A
2326:x
2316:1
2310:{
2305:=
2302:)
2299:x
2296:(
2291:A
2286:1
2280:,
2276:R
2269:)
2263:,
2260:X
2257:(
2254::
2249:A
2244:1
2218:,
2209:A
2190:,
2187:X
2181:A
2157:)
2151:,
2148:X
2145:(
2104:Y
2084:Y
2064:Y
2058:X
2055::
2050:n
2046:f
1999:.
1994:2
1981:3
1956:)
1951:4
1943:,
1938:1
1930:(
1910:Z
1904:X
1901::
1898:f
1892:g
1872:)
1867:4
1859:,
1856:Z
1853:(
1847:)
1842:3
1834:,
1831:Y
1828:(
1825::
1822:g
1802:)
1797:2
1789:,
1786:Y
1783:(
1777:)
1772:1
1764:,
1761:X
1758:(
1755::
1752:f
1730:.
1727:)
1722:3
1714:,
1711:Z
1708:(
1702:)
1697:1
1689:,
1686:X
1683:(
1680::
1677:f
1671:g
1651:)
1646:3
1638:,
1635:Z
1632:(
1626:)
1621:2
1613:,
1610:Y
1607:(
1604::
1601:g
1581:)
1576:2
1568:,
1565:Y
1562:(
1556:)
1551:1
1543:,
1540:X
1537:(
1534::
1531:f
1497:C
1490:X
1487::
1484:f
1463:,
1440:}
1431:f
1428:{
1425:,
1422:}
1413:f
1410:{
1407:,
1404:}
1395:f
1392:{
1372:.
1368:R
1341:}
1332:)
1329:x
1326:(
1323:f
1320::
1317:X
1311:x
1308:{
1305:=
1302:}
1293:f
1290:{
1270:f
1251:,
1247:R
1240:X
1237::
1234:f
1210:.
1206:C
1175:C
1168:B
1123:L
1101:,
1098:)
1092:C
1085:B
1079:,
1075:C
1071:(
1065:)
1060:L
1055:,
1051:R
1047:(
1044::
1041:f
1026:.
1010:,
1007:X
988:Y
960:)
957:T
954:,
951:Y
948:(
942:)
936:,
933:X
930:(
927::
924:f
900:)
897:T
894:,
891:Y
888:(
868:)
862:,
859:X
856:(
801:,
797:C
777:,
773:R
725:.
721:T
660:.
657:)
653:T
649:,
646:Y
643:(
637:)
631:,
628:X
625:(
619:f
599:Y
593:X
590::
587:f
563:)
560:f
557:(
534:,
525:)
522:f
519:(
494:.
485:}
482:E
476:)
473:x
470:(
467:f
461:X
455:x
452:{
446:)
443:E
440:(
435:1
428:f
407:T
400:E
360:f
340:E
319:T
312:E
292:Y
286:X
283::
280:f
260:.
256:T
193:Y
173:X
153:)
149:T
145:,
142:Y
139:(
119:)
113:,
110:X
107:(
20:)
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