1755:
Random variables induce pushforward measures. They map a probability space into a codomain space and endow that space with a probability measure defined by the pushforward. Furthermore, because random variables are functions (and hence total functions), the inverse image of the whole codomain is the
1740:
1287:
770:
1572:
432:
346:
252:
198:
832:
152:
106:
468:
521:
551:
919:
577:
1396:
1496:
1617:
875:
664:
1186:
603:
1592:
1441:
1362:
276:
1612:
1462:
1420:
2651:
2729:
2746:
1756:
whole domain, and the measure of the whole domain is 1, so the measure of the whole codomain is 1. This means that random variables can be composed
676:
1501:
2054:
1913:
1109:
on infinite-dimensional vector spaces are defined using the push-forward and the standard
Gaussian measure on the real line: a
354:
2569:
2400:
1940:
2561:
1813:
2807:
2347:
281:
2741:
1870:
2698:
2688:
2498:
2407:
2171:
203:
2027:
2736:
2683:
2577:
2483:
1782:
926:
157:
2602:
2582:
2546:
2470:
2190:
1906:
1786:
2724:
2503:
2465:
2417:
778:
111:
65:
2629:
2597:
2587:
2508:
2475:
2106:
2015:
1059:
is precisely its arc length (or, equivalently, the angle that it subtends at the centre of the circle.)
437:
490:
2646:
2551:
2327:
2255:
1760:
and they will always remain random variables and endow the codomain spaces with probability measures.
526:
2636:
2719:
2165:
2096:
2032:
880:
556:
2488:
2246:
2206:
1899:
1735:{\displaystyle \forall A\in \Sigma :\ \mu (A)=0\iff f_{*}\mu (A)=\mu {\big (}f^{-1}(A){\big )}=0}
1465:
1369:
1340:
1282:{\displaystyle f^{(n)}=\underbrace {f\circ f\circ \dots \circ f} _{n\mathrm {\,times} }:X\to X.}
2771:
2671:
2493:
2215:
2061:
1475:
1883:
848:
643:
2332:
2285:
2280:
2275:
2117:
2000:
1958:
1139:
582:
40:
2641:
2607:
2515:
2225:
2180:
2022:
1945:
1797:
1166:
1577:
1426:
1347:
261:
8:
2624:
2614:
2460:
2250:
1979:
1936:
1770:
933:
48:
2302:
2776:
2536:
2521:
2220:
2101:
2079:
1597:
1447:
1405:
2693:
2429:
2390:
2385:
2292:
2210:
1995:
1968:
1866:
1823:
1778:
1316:
1296:
1135:
2710:
2619:
2395:
2380:
2370:
2355:
2322:
2317:
2307:
2185:
2160:
1975:
1818:
1801:
1789:, and the maximal eigenvalue of the operator corresponds to the invariant measure.
1747:
1300:
1106:
950:
843:
475:
60:
44:
2786:
2766:
2541:
2439:
2434:
2412:
2270:
2235:
2155:
2049:
1862:
1774:
1117:
1094:
484:
1062:
The previous example extends nicely to give a natural "Lebesgue measure" on the
2676:
2531:
2526:
2337:
2312:
2265:
2195:
2175:
2135:
2125:
1922:
480:
20:
2801:
2781:
2444:
2365:
2360:
2260:
2230:
2200:
2150:
2145:
2140:
2130:
2044:
1963:
1879:
1785:. In finite spaces this operator typically satisfies the requirements of the
1110:
1090:
961:
2375:
2297:
2037:
1120:
1086:
2074:
982:
also denote the restriction of
Lebesgue measure to the interval [0, 2
921:. As with many induced mappings, this construction has the structure of a
765:{\displaystyle \int _{X_{2}}g\,d(f_{*}\mu )=\int _{X_{1}}g\circ f\,d\mu .}
2240:
1303:. It is often of interest in the study of such systems to find a measure
954:
1796:; as an operator on spaces of functions on measurable spaces, it is the
967:) may be defined using a push-forward construction and Lebesgue measure
2084:
1041:
937:
2066:
2010:
2005:
1097:
972:
2091:
1950:
1793:
842:
Pushforwards of measures allow to induce, from a function between
1891:
922:
1567:{\displaystyle \forall A\in \Sigma :\ \mu (A)=0\iff \nu (A)=0}
1067:
427:{\displaystyle f_{*}(\mu )(B)=\mu \left(f^{-1}(B)\right)}
1773:
can be pushed forward. The push-forward then becomes a
629:
is integrable with respect to the pushforward measure
1620:
1600:
1580:
1504:
1478:
1450:
1429:
1408:
1372:
1350:
1189:
883:
851:
781:
679:
646:
585:
559:
529:
493:
440:
357:
284:
264:
206:
160:
114:
68:
16:"Pushed forward" from one measurable space to another
1746:
Many natural probability distributions, such as the
39:) is obtained by transferring ("pushing forward") a
1734:
1606:
1586:
1566:
1490:
1472:, not necessarily equal to it. A pair of measures
1456:
1435:
1414:
1390:
1356:
1281:
913:
869:
826:
764:
658:
597:
571:
545:
515:
462:
426:
340:
270:
246:
192:
146:
100:
341:{\displaystyle f_{*}(\mu )\colon \Sigma _{2}\to }
2799:
1498:on the same space are equivalent if and only if
1073:. The previous example is a special case, since
936:, this property amounts to functoriality of the
670:. In that case, the integrals coincide, i.e.,
1907:
1721:
1692:
613:
487:. The pushforward measure is also denoted as
2652:RieszâMarkovâKakutani representation theorem
943:
877:, a function between the spaces of measures
666:is integrable with respect to the measure
2747:Vitale's random BrunnâMinkowski inequality
1914:
1900:
1661:
1657:
1545:
1541:
247:{\displaystyle \mu \colon \Sigma _{1}\to }
1245:
752:
700:
1856:
1842:
1750:, can be obtained via this construction.
1792:The adjoint to the push-forward is the
1343:for such a dynamical system: a measure
1044:measure" or "angle measure", since the
2800:
1885:Topics in Real and Functional Analysis
1878:
193:{\displaystyle f\colon X_{1}\to X_{2}}
1895:
1014:). The natural "Lebesgue measure" on
2760:Applications & related
998:be the natural bijection defined by
960:(here thought of as a subset of the
1814:Measure-preserving dynamical system
1764:
827:{\displaystyle X_{1}=f^{-1}(X_{2})}
147:{\displaystyle (X_{2},\Sigma _{2})}
101:{\displaystyle (X_{1},\Sigma _{1})}
13:
1921:
1630:
1621:
1514:
1505:
1382:
1258:
1255:
1252:
1249:
1246:
775:Note that in the previous formula
589:
563:
535:
448:
332:
308:
238:
214:
132:
86:
14:
2819:
1018:is then the push-forward measure
640:) if and only if the composition
463:{\displaystyle B\in \Sigma _{2}.}
2689:Lebesgue differentiation theorem
2570:Carathéodory's extension theorem
837:
516:{\displaystyle \mu \circ f^{-1}}
1783:Frobenius–Perron operator
1153:Consider a measurable function
618:Theorem: A measurable function
546:{\displaystyle f_{\sharp }\mu }
1857:Bogachev, Vladimir I. (2007),
1835:
1787:Frobenius–Perron theorem
1716:
1710:
1681:
1675:
1658:
1648:
1642:
1555:
1549:
1542:
1532:
1526:
1385:
1373:
1315:leaves unchanged, a so-called
1270:
1201:
1195:
908:
902:
896:
893:
887:
861:
821:
808:
720:
704:
416:
410:
383:
377:
374:
368:
335:
320:
317:
301:
295:
241:
226:
223:
177:
141:
115:
95:
69:
1:
1850:
1085:is, up to normalization, the
927:category of measurable spaces
608:
278:is defined to be the measure
54:
914:{\displaystyle M(X)\to M(Y)}
572:{\displaystyle f\sharp \mu }
7:
2742:PrĂ©kopaâLeindler inequality
1807:
1391:{\displaystyle (X,\Sigma )}
1081:. This Lebesgue measure on
10:
2824:
2684:Lebesgue's density theorem
614:Change of variable formula
2808:Measures (measure theory)
2759:
2737:MinkowskiâSteiner formula
2707:
2667:
2660:
2560:
2552:Projection-valued measure
2453:
2346:
2115:
1988:
1929:
1594:is quasi-invariant under
1491:{\displaystyle \mu ,\nu }
1146:is a Gaussian measure on
944:Examples and applications
2720:Isoperimetric inequality
2699:VitaliâHahnâSaks theorem
2028:Carathéodory's criterion
1829:
1468:to the original measure
1341:quasi-invariant measures
1040:) might also be called "
932:For the special case of
870:{\displaystyle f:X\to Y}
659:{\displaystyle g\circ f}
473:This definition applies
2725:BrunnâMinkowski theorem
2594:Decomposition theorems
1422:if the push-forward of
1130:if the push-forward of
1055:)-measure of an arc in
990: : [0, 2
598:{\displaystyle f\#\mu }
154:, a measurable mapping
2772:Descriptive set theory
2672:Disintegration theorem
2107:Universally measurable
1736:
1608:
1588:
1568:
1492:
1458:
1437:
1416:
1392:
1358:
1339:One can also consider
1283:
915:
871:
828:
766:
660:
599:
573:
547:
517:
464:
428:
342:
272:
248:
194:
148:
102:
2574:Convergence theorems
2033:Cylindrical Ï-algebra
1737:
1609:
1589:
1569:
1493:
1459:
1438:
1417:
1401:quasi-invariant under
1393:
1359:
1284:
1140:continuous dual space
916:
872:
829:
767:
661:
600:
574:
548:
518:
465:
429:
343:
273:
249:
195:
149:
103:
2642:Minkowski inequality
2516:Cylinder set measure
2401:Infinite-dimensional
2016:equivalence relation
1946:Lebesgue integration
1841:Sections 3.6â3.7 in
1798:composition operator
1618:
1598:
1587:{\displaystyle \mu }
1578:
1502:
1476:
1448:
1436:{\displaystyle \mu }
1427:
1406:
1370:
1357:{\displaystyle \mu }
1348:
1319:, i.e one for which
1187:
934:probability measures
881:
849:
779:
677:
644:
583:
557:
527:
491:
438:
355:
282:
271:{\displaystyle \mu }
262:
204:
158:
112:
66:
2637:Hölder's inequality
2499:of random variables
2461:Measurable function
2348:Particular measures
1937:Absolute continuity
1771:measurable function
49:measurable function
47:to another using a
25:pushforward measure
2777:Probability theory
2102:Transverse measure
2080:Non-measurable set
2062:Locally measurable
1732:
1604:
1584:
1564:
1488:
1454:
1433:
1412:
1388:
1354:
1279:
1263:
1238:
1006:) = exp(
911:
867:
824:
762:
656:
595:
569:
543:
513:
460:
424:
338:
268:
244:
190:
144:
98:
2795:
2794:
2755:
2754:
2484:almost everywhere
2430:Spherical measure
2328:Strictly positive
2256:Projection-valued
1996:Almost everywhere
1969:Probability space
1824:Optimal transport
1779:transfer operator
1638:
1607:{\displaystyle f}
1522:
1457:{\displaystyle f}
1415:{\displaystyle f}
1317:invariant measure
1297:iterated function
1211:
1209:
1136:linear functional
1107:Gaussian measures
844:measurable spaces
61:measurable spaces
2815:
2730:Milman's reverse
2713:
2711:Lebesgue measure
2665:
2664:
2069:
2055:infimum/supremum
1976:Measurable space
1916:
1909:
1902:
1893:
1892:
1888:
1875:
1845:
1839:
1819:Normalizing flow
1802:Koopman operator
1769:In general, any
1765:A generalization
1748:chi distribution
1741:
1739:
1738:
1733:
1725:
1724:
1709:
1708:
1696:
1695:
1671:
1670:
1636:
1613:
1611:
1610:
1605:
1593:
1591:
1590:
1585:
1573:
1571:
1570:
1565:
1520:
1497:
1495:
1494:
1489:
1463:
1461:
1460:
1455:
1442:
1440:
1439:
1434:
1421:
1419:
1418:
1413:
1397:
1395:
1394:
1389:
1363:
1361:
1360:
1355:
1301:dynamical system
1288:
1286:
1285:
1280:
1262:
1261:
1239:
1234:
1205:
1204:
1134:by any non-zero
951:Lebesgue measure
920:
918:
917:
912:
876:
874:
873:
868:
833:
831:
830:
825:
820:
819:
807:
806:
791:
790:
771:
769:
768:
763:
742:
741:
740:
739:
716:
715:
696:
695:
694:
693:
665:
663:
662:
657:
604:
602:
601:
596:
578:
576:
575:
570:
552:
550:
549:
544:
539:
538:
522:
520:
519:
514:
512:
511:
476:mutatis mutandis
469:
467:
466:
461:
456:
455:
433:
431:
430:
425:
423:
419:
409:
408:
367:
366:
347:
345:
344:
339:
316:
315:
294:
293:
277:
275:
274:
269:
253:
251:
250:
245:
222:
221:
199:
197:
196:
191:
189:
188:
176:
175:
153:
151:
150:
145:
140:
139:
127:
126:
107:
105:
104:
99:
94:
93:
81:
80:
45:measurable space
2823:
2822:
2818:
2817:
2816:
2814:
2813:
2812:
2798:
2797:
2796:
2791:
2787:Spectral theory
2767:Convex analysis
2751:
2708:
2703:
2656:
2556:
2504:in distribution
2449:
2342:
2172:Logarithmically
2111:
2067:
2050:Essential range
1984:
1925:
1920:
1873:
1863:Springer Verlag
1853:
1848:
1840:
1836:
1832:
1810:
1777:, known as the
1775:linear operator
1767:
1720:
1719:
1701:
1697:
1691:
1690:
1666:
1662:
1619:
1616:
1615:
1599:
1596:
1595:
1579:
1576:
1575:
1503:
1500:
1499:
1477:
1474:
1473:
1449:
1446:
1445:
1428:
1425:
1424:
1407:
1404:
1403:
1371:
1368:
1367:
1349:
1346:
1345:
1325:
1244:
1240:
1212:
1210:
1194:
1190:
1188:
1185:
1184:
1050:
1035:
1029:). The measure
1024:
946:
882:
879:
878:
850:
847:
846:
840:
815:
811:
799:
795:
786:
782:
780:
777:
776:
735:
731:
730:
726:
711:
707:
689:
685:
684:
680:
678:
675:
674:
645:
642:
641:
635:
628:
616:
611:
584:
581:
580:
558:
555:
554:
534:
530:
528:
525:
524:
504:
500:
492:
489:
488:
485:complex measure
451:
447:
439:
436:
435:
401:
397:
396:
392:
362:
358:
356:
353:
352:
311:
307:
289:
285:
283:
280:
279:
263:
260:
259:
217:
213:
205:
202:
201:
184:
180:
171:
167:
159:
156:
155:
135:
131:
122:
118:
113:
110:
109:
89:
85:
76:
72:
67:
64:
63:
57:
27:(also known as
17:
12:
11:
5:
2821:
2811:
2810:
2793:
2792:
2790:
2789:
2784:
2779:
2774:
2769:
2763:
2761:
2757:
2756:
2753:
2752:
2750:
2749:
2744:
2739:
2734:
2733:
2732:
2722:
2716:
2714:
2705:
2704:
2702:
2701:
2696:
2694:Sard's theorem
2691:
2686:
2681:
2680:
2679:
2677:Lifting theory
2668:
2662:
2658:
2657:
2655:
2654:
2649:
2644:
2639:
2634:
2633:
2632:
2630:FubiniâTonelli
2622:
2617:
2612:
2611:
2610:
2605:
2600:
2592:
2591:
2590:
2585:
2580:
2572:
2566:
2564:
2558:
2557:
2555:
2554:
2549:
2544:
2539:
2534:
2529:
2524:
2518:
2513:
2512:
2511:
2509:in probability
2506:
2496:
2491:
2486:
2480:
2479:
2478:
2473:
2468:
2457:
2455:
2451:
2450:
2448:
2447:
2442:
2437:
2432:
2427:
2422:
2421:
2420:
2410:
2405:
2404:
2403:
2393:
2388:
2383:
2378:
2373:
2368:
2363:
2358:
2352:
2350:
2344:
2343:
2341:
2340:
2335:
2330:
2325:
2320:
2315:
2310:
2305:
2300:
2295:
2290:
2289:
2288:
2283:
2278:
2268:
2263:
2258:
2253:
2243:
2238:
2233:
2228:
2223:
2218:
2216:Locally finite
2213:
2203:
2198:
2193:
2188:
2183:
2178:
2168:
2163:
2158:
2153:
2148:
2143:
2138:
2133:
2128:
2122:
2120:
2113:
2112:
2110:
2109:
2104:
2099:
2094:
2089:
2088:
2087:
2077:
2072:
2064:
2059:
2058:
2057:
2047:
2042:
2041:
2040:
2030:
2025:
2020:
2019:
2018:
2008:
2003:
1998:
1992:
1990:
1986:
1985:
1983:
1982:
1973:
1972:
1971:
1961:
1956:
1948:
1943:
1933:
1931:
1930:Basic concepts
1927:
1926:
1923:Measure theory
1919:
1918:
1911:
1904:
1896:
1890:
1889:
1880:Teschl, Gerald
1876:
1871:
1859:Measure Theory
1852:
1849:
1847:
1846:
1833:
1831:
1828:
1827:
1826:
1821:
1816:
1809:
1806:
1766:
1763:
1762:
1761:
1752:
1751:
1743:
1742:
1731:
1728:
1723:
1718:
1715:
1712:
1707:
1704:
1700:
1694:
1689:
1686:
1683:
1680:
1677:
1674:
1669:
1665:
1660:
1656:
1653:
1650:
1647:
1644:
1641:
1635:
1632:
1629:
1626:
1623:
1603:
1583:
1563:
1560:
1557:
1554:
1551:
1548:
1544:
1540:
1537:
1534:
1531:
1528:
1525:
1519:
1516:
1513:
1510:
1507:
1487:
1484:
1481:
1453:
1432:
1411:
1387:
1384:
1381:
1378:
1375:
1353:
1336:
1335:
1330:) =
1323:
1292:
1291:
1290:
1289:
1278:
1275:
1272:
1269:
1266:
1260:
1257:
1254:
1251:
1248:
1243:
1237:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1208:
1203:
1200:
1197:
1193:
1179:
1178:
1151:
1104:
1060:
1048:
1033:
1022:
994:) â
945:
942:
910:
907:
904:
901:
898:
895:
892:
889:
886:
866:
863:
860:
857:
854:
839:
836:
823:
818:
814:
810:
805:
802:
798:
794:
789:
785:
773:
772:
761:
758:
755:
751:
748:
745:
738:
734:
729:
725:
722:
719:
714:
710:
706:
703:
699:
692:
688:
683:
655:
652:
649:
633:
626:
615:
612:
610:
607:
594:
591:
588:
568:
565:
562:
542:
537:
533:
510:
507:
503:
499:
496:
471:
470:
459:
454:
450:
446:
443:
422:
418:
415:
412:
407:
404:
400:
395:
391:
388:
385:
382:
379:
376:
373:
370:
365:
361:
337:
334:
331:
328:
325:
322:
319:
314:
310:
306:
303:
300:
297:
292:
288:
267:
243:
240:
237:
234:
231:
228:
225:
220:
216:
212:
209:
200:and a measure
187:
183:
179:
174:
170:
166:
163:
143:
138:
134:
130:
125:
121:
117:
97:
92:
88:
84:
79:
75:
71:
56:
53:
21:measure theory
15:
9:
6:
4:
3:
2:
2820:
2809:
2806:
2805:
2803:
2788:
2785:
2783:
2782:Real analysis
2780:
2778:
2775:
2773:
2770:
2768:
2765:
2764:
2762:
2758:
2748:
2745:
2743:
2740:
2738:
2735:
2731:
2728:
2727:
2726:
2723:
2721:
2718:
2717:
2715:
2712:
2706:
2700:
2697:
2695:
2692:
2690:
2687:
2685:
2682:
2678:
2675:
2674:
2673:
2670:
2669:
2666:
2663:
2661:Other results
2659:
2653:
2650:
2648:
2647:RadonâNikodym
2645:
2643:
2640:
2638:
2635:
2631:
2628:
2627:
2626:
2623:
2621:
2620:Fatou's lemma
2618:
2616:
2613:
2609:
2606:
2604:
2601:
2599:
2596:
2595:
2593:
2589:
2586:
2584:
2581:
2579:
2576:
2575:
2573:
2571:
2568:
2567:
2565:
2563:
2559:
2553:
2550:
2548:
2545:
2543:
2540:
2538:
2535:
2533:
2530:
2528:
2525:
2523:
2519:
2517:
2514:
2510:
2507:
2505:
2502:
2501:
2500:
2497:
2495:
2492:
2490:
2487:
2485:
2482:Convergence:
2481:
2477:
2474:
2472:
2469:
2467:
2464:
2463:
2462:
2459:
2458:
2456:
2452:
2446:
2443:
2441:
2438:
2436:
2433:
2431:
2428:
2426:
2423:
2419:
2416:
2415:
2414:
2411:
2409:
2406:
2402:
2399:
2398:
2397:
2394:
2392:
2389:
2387:
2384:
2382:
2379:
2377:
2374:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2353:
2351:
2349:
2345:
2339:
2336:
2334:
2331:
2329:
2326:
2324:
2321:
2319:
2316:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2296:
2294:
2291:
2287:
2286:Outer regular
2284:
2282:
2281:Inner regular
2279:
2277:
2276:Borel regular
2274:
2273:
2272:
2269:
2267:
2264:
2262:
2259:
2257:
2254:
2252:
2248:
2244:
2242:
2239:
2237:
2234:
2232:
2229:
2227:
2224:
2222:
2219:
2217:
2214:
2212:
2208:
2204:
2202:
2199:
2197:
2194:
2192:
2189:
2187:
2184:
2182:
2179:
2177:
2173:
2169:
2167:
2164:
2162:
2159:
2157:
2154:
2152:
2149:
2147:
2144:
2142:
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2123:
2121:
2119:
2114:
2108:
2105:
2103:
2100:
2098:
2095:
2093:
2090:
2086:
2083:
2082:
2081:
2078:
2076:
2073:
2071:
2065:
2063:
2060:
2056:
2053:
2052:
2051:
2048:
2046:
2043:
2039:
2036:
2035:
2034:
2031:
2029:
2026:
2024:
2021:
2017:
2014:
2013:
2012:
2009:
2007:
2004:
2002:
1999:
1997:
1994:
1993:
1991:
1987:
1981:
1977:
1974:
1970:
1967:
1966:
1965:
1964:Measure space
1962:
1960:
1957:
1955:
1953:
1949:
1947:
1944:
1942:
1938:
1935:
1934:
1932:
1928:
1924:
1917:
1912:
1910:
1905:
1903:
1898:
1897:
1894:
1887:
1886:
1881:
1877:
1874:
1872:9783540345138
1868:
1864:
1860:
1855:
1854:
1844:
1843:Bogachev 2007
1838:
1834:
1825:
1822:
1820:
1817:
1815:
1812:
1811:
1805:
1803:
1799:
1795:
1790:
1788:
1784:
1780:
1776:
1772:
1759:
1754:
1753:
1749:
1745:
1744:
1729:
1726:
1713:
1705:
1702:
1698:
1687:
1684:
1678:
1672:
1667:
1663:
1654:
1651:
1645:
1639:
1633:
1627:
1624:
1601:
1581:
1561:
1558:
1552:
1546:
1538:
1535:
1529:
1523:
1517:
1511:
1508:
1485:
1482:
1479:
1471:
1467:
1451:
1443:
1430:
1409:
1402:
1398:
1379:
1376:
1364:
1351:
1342:
1338:
1337:
1333:
1329:
1322:
1318:
1314:
1311:that the map
1310:
1306:
1302:
1298:
1294:
1293:
1276:
1273:
1267:
1264:
1241:
1235:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1206:
1198:
1191:
1183:
1182:
1181:
1180:
1176:
1172:
1168:
1164:
1160:
1156:
1152:
1149:
1145:
1141:
1137:
1133:
1129:
1125:
1122:
1119:
1115:
1112:
1111:Borel measure
1108:
1105:
1102:
1099:
1096:
1092:
1088:
1084:
1080:
1077: =
1076:
1072:
1069:
1066:-dimensional
1065:
1061:
1058:
1054:
1047:
1043:
1039:
1032:
1028:
1021:
1017:
1013:
1009:
1005:
1001:
997:
993:
989:
985:
981:
977:
974:
970:
966:
963:
962:complex plane
959:
956:
952:
948:
947:
941:
939:
935:
930:
928:
924:
905:
899:
890:
884:
864:
858:
855:
852:
845:
838:Functoriality
835:
816:
812:
803:
800:
796:
792:
787:
783:
759:
756:
753:
749:
746:
743:
736:
732:
727:
723:
717:
712:
708:
701:
697:
690:
686:
681:
673:
672:
671:
669:
653:
650:
647:
639:
632:
625:
621:
606:
592:
586:
566:
560:
540:
531:
508:
505:
501:
497:
494:
486:
482:
478:
477:
457:
452:
444:
441:
420:
413:
405:
402:
398:
393:
389:
386:
380:
371:
363:
359:
351:
350:
349:
329:
326:
323:
312:
304:
298:
290:
286:
265:
257:
235:
232:
229:
218:
210:
207:
185:
181:
172:
168:
164:
161:
136:
128:
123:
119:
90:
82:
77:
73:
62:
52:
50:
46:
42:
38:
37:image measure
34:
30:
26:
22:
2562:Main results
2424:
2298:Set function
2226:Metric outer
2181:Decomposable
2038:Cylinder set
1951:
1884:
1858:
1837:
1791:
1768:
1758:ad infinitum
1757:
1469:
1423:
1400:
1366:
1344:
1331:
1327:
1320:
1312:
1308:
1304:
1174:
1173:with itself
1170:
1162:
1158:
1154:
1147:
1143:
1131:
1127:
1123:
1121:Banach space
1113:
1100:
1087:Haar measure
1082:
1078:
1074:
1070:
1063:
1056:
1052:
1045:
1037:
1030:
1026:
1019:
1015:
1011:
1007:
1003:
999:
995:
991:
987:
983:
979:
975:
968:
964:
957:
931:
841:
774:
667:
637:
630:
623:
619:
617:
474:
472:
255:
58:
36:
33:push-forward
32:
29:push forward
28:
24:
18:
2522:compact set
2489:of measures
2425:Pushforward
2418:Projections
2408:Logarithmic
2251:Probability
2241:Pre-measure
2023:Borel space
1941:of measures
1167:composition
955:unit circle
949:A natural "
256:pushforward
2494:in measure
2221:Maximising
2191:Equivalent
2085:Vitali set
1861:, Berlin:
1851:References
1466:equivalent
1464:is merely
1399:is called
1126:is called
1042:arc length
986:) and let
938:Giry monad
609:Properties
55:Definition
2608:Maharam's
2578:Dominated
2391:Intensity
2386:Hausdorff
2293:Saturated
2211:Invariant
2116:Types of
2075:Ï-algebra
2045:đ-system
2011:Borel set
2006:Baire set
1703:−
1688:μ
1673:μ
1668:∗
1659:⟺
1640:μ
1631:Σ
1628:∈
1622:∀
1582:μ
1547:ν
1543:⟺
1524:μ
1515:Σ
1512:∈
1506:∀
1486:ν
1480:μ
1431:μ
1383:Σ
1352:μ
1271:→
1236:⏟
1229:∘
1226:⋯
1223:∘
1217:∘
1118:separable
1098:Lie group
1095:connected
973:real line
953:" on the
925:, on the
897:→
862:→
801:−
757:μ
747:∘
728:∫
718:μ
713:∗
682:∫
651:∘
593:μ
590:#
567:μ
564:♯
541:μ
536:♯
506:−
498:∘
495:μ
449:Σ
445:∈
403:−
390:μ
372:μ
364:∗
348:given by
333:∞
318:→
309:Σ
305::
299:μ
291:∗
266:μ
239:∞
224:→
215:Σ
211::
208:μ
178:→
165::
133:Σ
87:Σ
43:from one
2802:Category
2625:Fubini's
2615:Egorov's
2583:Monotone
2542:variable
2520:Random:
2471:Strongly
2396:Lebesgue
2381:Harmonic
2371:Gaussian
2356:Counting
2323:Spectral
2318:Singular
2308:s-finite
2303:Ï-finite
2186:Discrete
2161:Complete
2118:Measures
2092:Null set
1980:function
1882:(2015),
1808:See also
1794:pullback
1299:forms a
1165:and the
1157: :
1128:Gaussian
1089:for the
2537:process
2532:measure
2527:element
2466:Bochner
2440:Trivial
2435:Tangent
2413:Product
2271:Regular
2249:)
2236:Perfect
2209:)
2174:)
2166:Content
2156:Complex
2097:Support
2070:-system
1959:Measure
1324:∗
1138:in the
1091:compact
971:on the
923:functor
41:measure
2603:Jordan
2588:Vitali
2547:vector
2476:Weakly
2338:Vector
2313:Signed
2266:Random
2207:Quasi-
2196:Finite
2176:Convex
2136:Banach
2126:Atomic
1954:spaces
1939:
1869:
1637:
1521:
1332:μ
1328:μ
1305:μ
1177:times:
1010:
978:. Let
481:signed
479:for a
254:, the
59:Given
2445:Young
2366:Euler
2361:Dirac
2333:Tight
2261:Radon
2231:Outer
2201:Inner
2151:Brown
2146:Borel
2141:Besov
2131:Baire
1830:Notes
1574:, so
1295:This
1116:on a
1068:torus
579:, or
2709:For
2598:Hahn
2454:Maps
2376:Haar
2247:Sub-
2001:Atom
1989:Sets
1867:ISBN
434:for
108:and
23:, a
1800:or
1781:or
1614:if
1444:by
1365:on
1307:on
1169:of
1142:to
622:on
483:or
258:of
35:or
19:In
2804::
1865:,
1804:.
1161:â
1093:,
940:.
929:.
834:.
605:.
553:,
523:,
51:.
31:,
2245:(
2205:(
2170:(
2068:Ï
1978:/
1952:L
1915:e
1908:t
1901:v
1730:0
1727:=
1722:)
1717:)
1714:A
1711:(
1706:1
1699:f
1693:(
1685:=
1682:)
1679:A
1676:(
1664:f
1655:0
1652:=
1649:)
1646:A
1643:(
1634::
1625:A
1602:f
1562:0
1559:=
1556:)
1553:A
1550:(
1539:0
1536:=
1533:)
1530:A
1527:(
1518::
1509:A
1483:,
1470:Ό
1452:f
1410:f
1386:)
1380:,
1377:X
1374:(
1334:.
1326:(
1321:f
1313:f
1309:X
1277:.
1274:X
1268:X
1265::
1259:s
1256:e
1253:m
1250:i
1247:t
1242:n
1232:f
1220:f
1214:f
1207:=
1202:)
1199:n
1196:(
1192:f
1175:n
1171:f
1163:X
1159:X
1155:f
1150:.
1148:R
1144:X
1132:Îł
1124:X
1114:Îł
1103:.
1101:T
1083:T
1079:T
1075:S
1071:T
1064:n
1057:S
1053:λ
1051:(
1049:â
1046:f
1038:λ
1036:(
1034:â
1031:f
1027:λ
1025:(
1023:â
1020:f
1016:S
1012:t
1008:i
1004:t
1002:(
1000:f
996:S
992:Ï
988:f
984:Ï
980:λ
976:R
969:λ
965:C
958:S
909:)
906:Y
903:(
900:M
894:)
891:X
888:(
885:M
865:Y
859:X
856::
853:f
822:)
817:2
813:X
809:(
804:1
797:f
793:=
788:1
784:X
760:.
754:d
750:f
744:g
737:1
733:X
724:=
721:)
709:f
705:(
702:d
698:g
691:2
687:X
668:Ό
654:f
648:g
638:Ό
636:(
634:â
631:f
627:2
624:X
620:g
587:f
561:f
532:f
509:1
502:f
458:.
453:2
442:B
421:)
417:)
414:B
411:(
406:1
399:f
394:(
387:=
384:)
381:B
378:(
375:)
369:(
360:f
336:]
330:+
327:,
324:0
321:[
313:2
302:)
296:(
287:f
242:]
236:+
233:,
230:0
227:[
219:1
186:2
182:X
173:1
169:X
162:f
142:)
137:2
129:,
124:2
120:X
116:(
96:)
91:1
83:,
78:1
74:X
70:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.