1766:
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
1751:
1298:
781:
1583:
443:
357:
263:
209:
843:
163:
117:
479:
532:
562:
930:
588:
1407:
1507:
1628:
886:
675:
1197:
614:
1603:
1452:
1373:
287:
1623:
1473:
1431:
2662:
2740:
2757:
1767:
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
687:
1512:
2065:
1924:
1120:
on infinite-dimensional vector spaces are defined using the push-forward and the standard
Gaussian measure on the real line: a
365:
2580:
2411:
1951:
17:
2572:
1824:
2818:
2358:
292:
2752:
1881:
2709:
2699:
2509:
2418:
2182:
214:
2038:
2747:
2694:
2588:
2494:
1793:
937:
168:
2613:
2593:
2557:
2481:
2201:
1917:
1797:
2735:
2514:
2476:
2428:
789:
122:
76:
2640:
2608:
2598:
2519:
2486:
2117:
2026:
1070:
is precisely its arc length (or, equivalently, the angle that it subtends at the centre of the circle.)
448:
501:
2657:
2562:
2338:
2266:
1771:
and they will always remain random variables and endow the codomain spaces with probability measures.
537:
2647:
2730:
2176:
2107:
2043:
891:
567:
2499:
2257:
2217:
1910:
1746:{\displaystyle \forall A\in \Sigma :\ \mu (A)=0\iff f_{*}\mu (A)=\mu {\big (}f^{-1}(A){\big )}=0}
1476:
1380:
1351:
1293:{\displaystyle f^{(n)}=\underbrace {f\circ f\circ \dots \circ f} _{n\mathrm {\,times} }:X\to X.}
2782:
2682:
2504:
2226:
2072:
1486:
1894:
859:
654:
2343:
2296:
2291:
2286:
2128:
2011:
1969:
1150:
593:
51:
2652:
2618:
2526:
2236:
2191:
2033:
1956:
1808:
1177:
1588:
1437:
1358:
272:
8:
2635:
2625:
2471:
2261:
1990:
1947:
1781:
944:
59:
2313:
2787:
2547:
2532:
2231:
2112:
2090:
1608:
1458:
1416:
2704:
2440:
2401:
2396:
2303:
2221:
2006:
1979:
1877:
1834:
1789:
1327:
1307:
1146:
2721:
2630:
2406:
2391:
2381:
2366:
2333:
2328:
2318:
2196:
2171:
1986:
1829:
1812:
1800:, and the maximal eigenvalue of the operator corresponds to the invariant measure.
1758:
1311:
1117:
961:
854:
486:
71:
55:
2797:
2777:
2552:
2450:
2445:
2423:
2281:
2246:
2166:
2060:
1873:
1785:
1128:
1105:
495:
1073:
The previous example extends nicely to give a natural "Lebesgue measure" on the
2687:
2542:
2537:
2348:
2323:
2276:
2206:
2186:
2146:
2136:
1933:
491:
31:
2812:
2792:
2455:
2376:
2371:
2271:
2241:
2211:
2161:
2156:
2151:
2141:
2055:
1974:
1890:
1796:. In finite spaces this operator typically satisfies the requirements of the
1121:
1101:
972:
2386:
2308:
2048:
1131:
1097:
2085:
993:
also denote the restriction of
Lebesgue measure to the interval [0, 2
932:. As with many induced mappings, this construction has the structure of a
776:{\displaystyle \int _{X_{2}}g\,d(f_{*}\mu )=\int _{X_{1}}g\circ f\,d\mu .}
2251:
1314:. It is often of interest in the study of such systems to find a measure
965:
1807:; as an operator on spaces of functions on measurable spaces, it is the
978:) may be defined using a push-forward construction and Lebesgue measure
2095:
1052:
948:
2077:
2021:
2016:
1108:
983:
2102:
1961:
1804:
853:
Pushforwards of measures allow to induce, from a function between
1902:
933:
1578:{\displaystyle \forall A\in \Sigma :\ \mu (A)=0\iff \nu (A)=0}
1078:
438:{\displaystyle f_{*}(\mu )(B)=\mu \left(f^{-1}(B)\right)}
1784:
can be pushed forward. The push-forward then becomes a
640:
is integrable with respect to the pushforward measure
1631:
1611:
1591:
1515:
1489:
1461:
1440:
1419:
1383:
1361:
1200:
894:
862:
792:
690:
657:
596:
570:
540:
504:
451:
368:
295:
275:
217:
171:
125:
79:
27:"Pushed forward" from one measurable space to another
1757:
Many natural probability distributions, such as the
50:) is obtained by transferring ("pushing forward") a
1745:
1617:
1597:
1577:
1501:
1483:, not necessarily equal to it. A pair of measures
1467:
1446:
1425:
1401:
1367:
1292:
924:
880:
837:
775:
669:
608:
582:
556:
526:
473:
437:
351:
281:
257:
203:
157:
111:
352:{\displaystyle f_{*}(\mu )\colon \Sigma _{2}\to }
2810:
1509:on the same space are equivalent if and only if
1084:. The previous example is a special case, since
947:, this property amounts to functoriality of the
681:. In that case, the integrals coincide, i.e.,
1918:
1732:
1703:
624:
498:. The pushforward measure is also denoted as
2663:RieszâMarkovâKakutani representation theorem
954:
888:, a function between the spaces of measures
677:is integrable with respect to the measure
2758:Vitale's random BrunnâMinkowski inequality
1925:
1911:
1672:
1668:
1556:
1552:
258:{\displaystyle \mu \colon \Sigma _{1}\to }
1256:
763:
711:
1867:
1853:
1761:, can be obtained via this construction.
1803:The adjoint to the push-forward is the
1354:for such a dynamical system: a measure
1055:measure" or "angle measure", since the
14:
2811:
1896:Topics in Real and Functional Analysis
1889:
204:{\displaystyle f\colon X_{1}\to X_{2}}
1906:
1025:). The natural "Lebesgue measure" on
2771:Applications & related
1009:be the natural bijection defined by
971:(here thought of as a subset of the
1825:Measure-preserving dynamical system
1775:
838:{\displaystyle X_{1}=f^{-1}(X_{2})}
158:{\displaystyle (X_{2},\Sigma _{2})}
112:{\displaystyle (X_{1},\Sigma _{1})}
24:
1932:
1641:
1632:
1525:
1516:
1393:
1269:
1266:
1263:
1260:
1257:
786:Note that in the previous formula
600:
574:
546:
459:
343:
319:
249:
225:
143:
97:
25:
2830:
1029:is then the push-forward measure
651:) if and only if the composition
474:{\displaystyle B\in \Sigma _{2}.}
2700:Lebesgue differentiation theorem
2581:Carathéodory's extension theorem
848:
527:{\displaystyle \mu \circ f^{-1}}
1794:Frobenius–Perron operator
1164:Consider a measurable function
629:Theorem: A measurable function
557:{\displaystyle f_{\sharp }\mu }
1868:Bogachev, Vladimir I. (2007),
1846:
1798:Frobenius–Perron theorem
1727:
1721:
1692:
1686:
1669:
1659:
1653:
1566:
1560:
1553:
1543:
1537:
1396:
1384:
1326:leaves unchanged, a so-called
1281:
1212:
1206:
919:
913:
907:
904:
898:
872:
832:
819:
731:
715:
427:
421:
394:
388:
385:
379:
346:
331:
328:
312:
306:
252:
237:
234:
188:
152:
126:
106:
80:
13:
1:
1861:
1096:is, up to normalization, the
938:category of measurable spaces
619:
289:is defined to be the measure
65:
925:{\displaystyle M(X)\to M(Y)}
583:{\displaystyle f\sharp \mu }
7:
2753:PrĂ©kopaâLeindler inequality
1818:
1402:{\displaystyle (X,\Sigma )}
1092:. This Lebesgue measure on
10:
2835:
2695:Lebesgue's density theorem
625:Change of variable formula
2819:Measures (measure theory)
2770:
2748:MinkowskiâSteiner formula
2718:
2678:
2671:
2571:
2563:Projection-valued measure
2464:
2357:
2126:
1999:
1940:
1605:is quasi-invariant under
1502:{\displaystyle \mu ,\nu }
1157:is a Gaussian measure on
955:Examples and applications
2731:Isoperimetric inequality
2710:VitaliâHahnâSaks theorem
2039:Carathéodory's criterion
1840:
1479:to the original measure
1352:quasi-invariant measures
1051:) might also be called "
943:For the special case of
881:{\displaystyle f:X\to Y}
670:{\displaystyle g\circ f}
484:This definition applies
2736:BrunnâMinkowski theorem
2605:Decomposition theorems
1433:if the push-forward of
1141:if the push-forward of
1066:)-measure of an arc in
1001: : [0, 2
609:{\displaystyle f\#\mu }
165:, a measurable mapping
2783:Descriptive set theory
2683:Disintegration theorem
2118:Universally measurable
1747:
1619:
1599:
1579:
1503:
1469:
1448:
1427:
1403:
1369:
1350:One can also consider
1294:
926:
882:
839:
777:
671:
610:
584:
558:
528:
475:
439:
353:
283:
259:
205:
159:
113:
2585:Convergence theorems
2044:Cylindrical Ï-algebra
1748:
1620:
1600:
1580:
1504:
1470:
1449:
1428:
1412:quasi-invariant under
1404:
1370:
1295:
1151:continuous dual space
927:
883:
840:
778:
672:
611:
585:
559:
529:
476:
440:
354:
284:
260:
206:
160:
114:
2653:Minkowski inequality
2527:Cylinder set measure
2412:Infinite-dimensional
2027:equivalence relation
1957:Lebesgue integration
1852:Sections 3.6â3.7 in
1809:composition operator
1629:
1609:
1598:{\displaystyle \mu }
1589:
1513:
1487:
1459:
1447:{\displaystyle \mu }
1438:
1417:
1381:
1368:{\displaystyle \mu }
1359:
1330:, i.e one for which
1198:
945:probability measures
892:
860:
790:
688:
655:
594:
568:
538:
502:
449:
366:
293:
282:{\displaystyle \mu }
273:
215:
169:
123:
77:
18:Push-forward measure
2648:Hölder's inequality
2510:of random variables
2472:Measurable function
2359:Particular measures
1948:Absolute continuity
1782:measurable function
60:measurable function
58:to another using a
36:pushforward measure
2788:Probability theory
2113:Transverse measure
2091:Non-measurable set
2073:Locally measurable
1743:
1615:
1595:
1575:
1499:
1465:
1444:
1423:
1399:
1365:
1290:
1274:
1249:
1017:) = exp(
922:
878:
835:
773:
667:
606:
580:
554:
524:
471:
435:
349:
279:
255:
201:
155:
109:
2806:
2805:
2766:
2765:
2495:almost everywhere
2441:Spherical measure
2339:Strictly positive
2267:Projection-valued
2007:Almost everywhere
1980:Probability space
1835:Optimal transport
1790:transfer operator
1649:
1618:{\displaystyle f}
1533:
1468:{\displaystyle f}
1426:{\displaystyle f}
1328:invariant measure
1308:iterated function
1222:
1220:
1147:linear functional
1118:Gaussian measures
855:measurable spaces
72:measurable spaces
16:(Redirected from
2826:
2741:Milman's reverse
2724:
2722:Lebesgue measure
2676:
2675:
2080:
2066:infimum/supremum
1987:Measurable space
1927:
1920:
1913:
1904:
1903:
1899:
1886:
1856:
1850:
1830:Normalizing flow
1813:Koopman operator
1780:In general, any
1776:A generalization
1759:chi distribution
1752:
1750:
1749:
1744:
1736:
1735:
1720:
1719:
1707:
1706:
1682:
1681:
1647:
1624:
1622:
1621:
1616:
1604:
1602:
1601:
1596:
1584:
1582:
1581:
1576:
1531:
1508:
1506:
1505:
1500:
1474:
1472:
1471:
1466:
1453:
1451:
1450:
1445:
1432:
1430:
1429:
1424:
1408:
1406:
1405:
1400:
1374:
1372:
1371:
1366:
1312:dynamical system
1299:
1297:
1296:
1291:
1273:
1272:
1250:
1245:
1216:
1215:
1145:by any non-zero
962:Lebesgue measure
931:
929:
928:
923:
887:
885:
884:
879:
844:
842:
841:
836:
831:
830:
818:
817:
802:
801:
782:
780:
779:
774:
753:
752:
751:
750:
727:
726:
707:
706:
705:
704:
676:
674:
673:
668:
615:
613:
612:
607:
589:
587:
586:
581:
563:
561:
560:
555:
550:
549:
533:
531:
530:
525:
523:
522:
487:mutatis mutandis
480:
478:
477:
472:
467:
466:
444:
442:
441:
436:
434:
430:
420:
419:
378:
377:
358:
356:
355:
350:
327:
326:
305:
304:
288:
286:
285:
280:
264:
262:
261:
256:
233:
232:
210:
208:
207:
202:
200:
199:
187:
186:
164:
162:
161:
156:
151:
150:
138:
137:
118:
116:
115:
110:
105:
104:
92:
91:
56:measurable space
21:
2834:
2833:
2829:
2828:
2827:
2825:
2824:
2823:
2809:
2808:
2807:
2802:
2798:Spectral theory
2778:Convex analysis
2762:
2719:
2714:
2667:
2567:
2515:in distribution
2460:
2353:
2183:Logarithmically
2122:
2078:
2061:Essential range
1995:
1936:
1931:
1884:
1874:Springer Verlag
1864:
1859:
1851:
1847:
1843:
1821:
1788:, known as the
1786:linear operator
1778:
1731:
1730:
1712:
1708:
1702:
1701:
1677:
1673:
1630:
1627:
1626:
1610:
1607:
1606:
1590:
1587:
1586:
1514:
1511:
1510:
1488:
1485:
1484:
1460:
1457:
1456:
1439:
1436:
1435:
1418:
1415:
1414:
1382:
1379:
1378:
1360:
1357:
1356:
1336:
1255:
1251:
1223:
1221:
1205:
1201:
1199:
1196:
1195:
1061:
1046:
1040:). The measure
1035:
957:
893:
890:
889:
861:
858:
857:
851:
826:
822:
810:
806:
797:
793:
791:
788:
787:
746:
742:
741:
737:
722:
718:
700:
696:
695:
691:
689:
686:
685:
656:
653:
652:
646:
639:
627:
622:
595:
592:
591:
569:
566:
565:
545:
541:
539:
536:
535:
515:
511:
503:
500:
499:
496:complex measure
462:
458:
450:
447:
446:
412:
408:
407:
403:
373:
369:
367:
364:
363:
322:
318:
300:
296:
294:
291:
290:
274:
271:
270:
228:
224:
216:
213:
212:
195:
191:
182:
178:
170:
167:
166:
146:
142:
133:
129:
124:
121:
120:
100:
96:
87:
83:
78:
75:
74:
68:
38:(also known as
28:
23:
22:
15:
12:
11:
5:
2832:
2822:
2821:
2804:
2803:
2801:
2800:
2795:
2790:
2785:
2780:
2774:
2772:
2768:
2767:
2764:
2763:
2761:
2760:
2755:
2750:
2745:
2744:
2743:
2733:
2727:
2725:
2716:
2715:
2713:
2712:
2707:
2705:Sard's theorem
2702:
2697:
2692:
2691:
2690:
2688:Lifting theory
2679:
2673:
2669:
2668:
2666:
2665:
2660:
2655:
2650:
2645:
2644:
2643:
2641:FubiniâTonelli
2633:
2628:
2623:
2622:
2621:
2616:
2611:
2603:
2602:
2601:
2596:
2591:
2583:
2577:
2575:
2569:
2568:
2566:
2565:
2560:
2555:
2550:
2545:
2540:
2535:
2529:
2524:
2523:
2522:
2520:in probability
2517:
2507:
2502:
2497:
2491:
2490:
2489:
2484:
2479:
2468:
2466:
2462:
2461:
2459:
2458:
2453:
2448:
2443:
2438:
2433:
2432:
2431:
2421:
2416:
2415:
2414:
2404:
2399:
2394:
2389:
2384:
2379:
2374:
2369:
2363:
2361:
2355:
2354:
2352:
2351:
2346:
2341:
2336:
2331:
2326:
2321:
2316:
2311:
2306:
2301:
2300:
2299:
2294:
2289:
2279:
2274:
2269:
2264:
2254:
2249:
2244:
2239:
2234:
2229:
2227:Locally finite
2224:
2214:
2209:
2204:
2199:
2194:
2189:
2179:
2174:
2169:
2164:
2159:
2154:
2149:
2144:
2139:
2133:
2131:
2124:
2123:
2121:
2120:
2115:
2110:
2105:
2100:
2099:
2098:
2088:
2083:
2075:
2070:
2069:
2068:
2058:
2053:
2052:
2051:
2041:
2036:
2031:
2030:
2029:
2019:
2014:
2009:
2003:
2001:
1997:
1996:
1994:
1993:
1984:
1983:
1982:
1972:
1967:
1959:
1954:
1944:
1942:
1941:Basic concepts
1938:
1937:
1934:Measure theory
1930:
1929:
1922:
1915:
1907:
1901:
1900:
1891:Teschl, Gerald
1887:
1882:
1870:Measure Theory
1863:
1860:
1858:
1857:
1844:
1842:
1839:
1838:
1837:
1832:
1827:
1820:
1817:
1777:
1774:
1773:
1772:
1763:
1762:
1754:
1753:
1742:
1739:
1734:
1729:
1726:
1723:
1718:
1715:
1711:
1705:
1700:
1697:
1694:
1691:
1688:
1685:
1680:
1676:
1671:
1667:
1664:
1661:
1658:
1655:
1652:
1646:
1643:
1640:
1637:
1634:
1614:
1594:
1574:
1571:
1568:
1565:
1562:
1559:
1555:
1551:
1548:
1545:
1542:
1539:
1536:
1530:
1527:
1524:
1521:
1518:
1498:
1495:
1492:
1464:
1443:
1422:
1398:
1395:
1392:
1389:
1386:
1364:
1347:
1346:
1341:) =
1334:
1303:
1302:
1301:
1300:
1289:
1286:
1283:
1280:
1277:
1271:
1268:
1265:
1262:
1259:
1254:
1248:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1219:
1214:
1211:
1208:
1204:
1190:
1189:
1162:
1115:
1071:
1059:
1044:
1033:
1005:) â
956:
953:
921:
918:
915:
912:
909:
906:
903:
900:
897:
877:
874:
871:
868:
865:
850:
847:
834:
829:
825:
821:
816:
813:
809:
805:
800:
796:
784:
783:
772:
769:
766:
762:
759:
756:
749:
745:
740:
736:
733:
730:
725:
721:
717:
714:
710:
703:
699:
694:
666:
663:
660:
644:
637:
626:
623:
621:
618:
605:
602:
599:
579:
576:
573:
553:
548:
544:
521:
518:
514:
510:
507:
482:
481:
470:
465:
461:
457:
454:
433:
429:
426:
423:
418:
415:
411:
406:
402:
399:
396:
393:
390:
387:
384:
381:
376:
372:
348:
345:
342:
339:
336:
333:
330:
325:
321:
317:
314:
311:
308:
303:
299:
278:
254:
251:
248:
245:
242:
239:
236:
231:
227:
223:
220:
211:and a measure
198:
194:
190:
185:
181:
177:
174:
154:
149:
145:
141:
136:
132:
128:
108:
103:
99:
95:
90:
86:
82:
67:
64:
32:measure theory
26:
9:
6:
4:
3:
2:
2831:
2820:
2817:
2816:
2814:
2799:
2796:
2794:
2793:Real analysis
2791:
2789:
2786:
2784:
2781:
2779:
2776:
2775:
2773:
2769:
2759:
2756:
2754:
2751:
2749:
2746:
2742:
2739:
2738:
2737:
2734:
2732:
2729:
2728:
2726:
2723:
2717:
2711:
2708:
2706:
2703:
2701:
2698:
2696:
2693:
2689:
2686:
2685:
2684:
2681:
2680:
2677:
2674:
2672:Other results
2670:
2664:
2661:
2659:
2658:RadonâNikodym
2656:
2654:
2651:
2649:
2646:
2642:
2639:
2638:
2637:
2634:
2632:
2631:Fatou's lemma
2629:
2627:
2624:
2620:
2617:
2615:
2612:
2610:
2607:
2606:
2604:
2600:
2597:
2595:
2592:
2590:
2587:
2586:
2584:
2582:
2579:
2578:
2576:
2574:
2570:
2564:
2561:
2559:
2556:
2554:
2551:
2549:
2546:
2544:
2541:
2539:
2536:
2534:
2530:
2528:
2525:
2521:
2518:
2516:
2513:
2512:
2511:
2508:
2506:
2503:
2501:
2498:
2496:
2493:Convergence:
2492:
2488:
2485:
2483:
2480:
2478:
2475:
2474:
2473:
2470:
2469:
2467:
2463:
2457:
2454:
2452:
2449:
2447:
2444:
2442:
2439:
2437:
2434:
2430:
2427:
2426:
2425:
2422:
2420:
2417:
2413:
2410:
2409:
2408:
2405:
2403:
2400:
2398:
2395:
2393:
2390:
2388:
2385:
2383:
2380:
2378:
2375:
2373:
2370:
2368:
2365:
2364:
2362:
2360:
2356:
2350:
2347:
2345:
2342:
2340:
2337:
2335:
2332:
2330:
2327:
2325:
2322:
2320:
2317:
2315:
2312:
2310:
2307:
2305:
2302:
2298:
2297:Outer regular
2295:
2293:
2292:Inner regular
2290:
2288:
2287:Borel regular
2285:
2284:
2283:
2280:
2278:
2275:
2273:
2270:
2268:
2265:
2263:
2259:
2255:
2253:
2250:
2248:
2245:
2243:
2240:
2238:
2235:
2233:
2230:
2228:
2225:
2223:
2219:
2215:
2213:
2210:
2208:
2205:
2203:
2200:
2198:
2195:
2193:
2190:
2188:
2184:
2180:
2178:
2175:
2173:
2170:
2168:
2165:
2163:
2160:
2158:
2155:
2153:
2150:
2148:
2145:
2143:
2140:
2138:
2135:
2134:
2132:
2130:
2125:
2119:
2116:
2114:
2111:
2109:
2106:
2104:
2101:
2097:
2094:
2093:
2092:
2089:
2087:
2084:
2082:
2076:
2074:
2071:
2067:
2064:
2063:
2062:
2059:
2057:
2054:
2050:
2047:
2046:
2045:
2042:
2040:
2037:
2035:
2032:
2028:
2025:
2024:
2023:
2020:
2018:
2015:
2013:
2010:
2008:
2005:
2004:
2002:
1998:
1992:
1988:
1985:
1981:
1978:
1977:
1976:
1975:Measure space
1973:
1971:
1968:
1966:
1964:
1960:
1958:
1955:
1953:
1949:
1946:
1945:
1943:
1939:
1935:
1928:
1923:
1921:
1916:
1914:
1909:
1908:
1905:
1898:
1897:
1892:
1888:
1885:
1883:9783540345138
1879:
1875:
1871:
1866:
1865:
1855:
1854:Bogachev 2007
1849:
1845:
1836:
1833:
1831:
1828:
1826:
1823:
1822:
1816:
1814:
1810:
1806:
1801:
1799:
1795:
1791:
1787:
1783:
1770:
1765:
1764:
1760:
1756:
1755:
1740:
1737:
1724:
1716:
1713:
1709:
1698:
1695:
1689:
1683:
1678:
1674:
1665:
1662:
1656:
1650:
1644:
1638:
1635:
1612:
1592:
1572:
1569:
1563:
1557:
1549:
1546:
1540:
1534:
1528:
1522:
1519:
1496:
1493:
1490:
1482:
1478:
1462:
1454:
1441:
1420:
1413:
1409:
1390:
1387:
1375:
1362:
1353:
1349:
1348:
1344:
1340:
1333:
1329:
1325:
1322:that the map
1321:
1317:
1313:
1309:
1305:
1304:
1287:
1284:
1278:
1275:
1252:
1246:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1217:
1209:
1202:
1194:
1193:
1192:
1191:
1187:
1183:
1179:
1175:
1171:
1167:
1163:
1160:
1156:
1152:
1148:
1144:
1140:
1136:
1133:
1130:
1126:
1123:
1122:Borel measure
1119:
1116:
1113:
1110:
1107:
1103:
1099:
1095:
1091:
1088: =
1087:
1083:
1080:
1077:-dimensional
1076:
1072:
1069:
1065:
1058:
1054:
1050:
1043:
1039:
1032:
1028:
1024:
1020:
1016:
1012:
1008:
1004:
1000:
996:
992:
988:
985:
981:
977:
974:
973:complex plane
970:
967:
963:
959:
958:
952:
950:
946:
941:
939:
935:
916:
910:
901:
895:
875:
869:
866:
863:
856:
849:Functoriality
846:
827:
823:
814:
811:
807:
803:
798:
794:
770:
767:
764:
760:
757:
754:
747:
743:
738:
734:
728:
723:
719:
712:
708:
701:
697:
692:
684:
683:
682:
680:
664:
661:
658:
650:
643:
636:
632:
617:
603:
597:
577:
571:
551:
542:
519:
516:
512:
508:
505:
497:
493:
489:
488:
468:
463:
455:
452:
431:
424:
416:
413:
409:
404:
400:
397:
391:
382:
374:
370:
362:
361:
360:
340:
337:
334:
323:
315:
309:
301:
297:
276:
268:
246:
243:
240:
229:
221:
218:
196:
192:
183:
179:
175:
172:
147:
139:
134:
130:
101:
93:
88:
84:
73:
63:
61:
57:
53:
49:
48:image measure
45:
41:
37:
33:
19:
2573:Main results
2435:
2309:Set function
2237:Metric outer
2192:Decomposable
2049:Cylinder set
1962:
1895:
1869:
1848:
1802:
1779:
1769:ad infinitum
1768:
1480:
1434:
1411:
1377:
1355:
1342:
1338:
1331:
1323:
1319:
1315:
1185:
1184:with itself
1181:
1173:
1169:
1165:
1158:
1154:
1142:
1138:
1134:
1132:Banach space
1124:
1111:
1098:Haar measure
1093:
1089:
1085:
1081:
1074:
1067:
1063:
1056:
1048:
1041:
1037:
1030:
1026:
1022:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
979:
975:
968:
942:
852:
785:
678:
648:
641:
634:
630:
628:
485:
483:
266:
69:
47:
44:push-forward
43:
40:push forward
39:
35:
29:
2533:compact set
2500:of measures
2436:Pushforward
2429:Projections
2419:Logarithmic
2262:Probability
2252:Pre-measure
2034:Borel space
1952:of measures
1178:composition
966:unit circle
960:A natural "
267:pushforward
2505:in measure
2232:Maximising
2202:Equivalent
2096:Vitali set
1872:, Berlin:
1862:References
1477:equivalent
1475:is merely
1410:is called
1137:is called
1053:arc length
997:) and let
949:Giry monad
620:Properties
66:Definition
2619:Maharam's
2589:Dominated
2402:Intensity
2397:Hausdorff
2304:Saturated
2222:Invariant
2127:Types of
2086:Ï-algebra
2056:đ-system
2022:Borel set
2017:Baire set
1714:−
1699:μ
1684:μ
1679:∗
1670:⟺
1651:μ
1642:Σ
1639:∈
1633:∀
1593:μ
1558:ν
1554:⟺
1535:μ
1526:Σ
1523:∈
1517:∀
1497:ν
1491:μ
1442:μ
1394:Σ
1363:μ
1282:→
1247:⏟
1240:∘
1237:⋯
1234:∘
1228:∘
1129:separable
1109:Lie group
1106:connected
984:real line
964:" on the
936:, on the
908:→
873:→
812:−
768:μ
758:∘
739:∫
729:μ
724:∗
693:∫
662:∘
604:μ
601:#
578:μ
575:♯
552:μ
547:♯
517:−
509:∘
506:μ
460:Σ
456:∈
414:−
401:μ
383:μ
375:∗
359:given by
344:∞
329:→
320:Σ
316::
310:μ
302:∗
277:μ
250:∞
235:→
226:Σ
222::
219:μ
189:→
176::
144:Σ
98:Σ
54:from one
2813:Category
2636:Fubini's
2626:Egorov's
2594:Monotone
2553:variable
2531:Random:
2482:Strongly
2407:Lebesgue
2392:Harmonic
2382:Gaussian
2367:Counting
2334:Spectral
2329:Singular
2319:s-finite
2314:Ï-finite
2197:Discrete
2172:Complete
2129:Measures
2103:Null set
1991:function
1893:(2015),
1819:See also
1805:pullback
1310:forms a
1176:and the
1168: :
1139:Gaussian
1100:for the
2548:process
2543:measure
2538:element
2477:Bochner
2451:Trivial
2446:Tangent
2424:Product
2282:Regular
2260:)
2247:Perfect
2220:)
2185:)
2177:Content
2167:Complex
2108:Support
2081:-system
1970:Measure
1335:∗
1149:in the
1102:compact
982:on the
934:functor
52:measure
2614:Jordan
2599:Vitali
2558:vector
2487:Weakly
2349:Vector
2324:Signed
2277:Random
2218:Quasi-
2207:Finite
2187:Convex
2147:Banach
2137:Atomic
1965:spaces
1950:
1880:
1648:
1532:
1343:μ
1339:μ
1316:μ
1188:times:
1021:
989:. Let
492:signed
490:for a
265:, the
70:Given
2456:Young
2377:Euler
2372:Dirac
2344:Tight
2272:Radon
2242:Outer
2212:Inner
2162:Brown
2157:Borel
2152:Besov
2142:Baire
1841:Notes
1585:, so
1306:This
1127:on a
1079:torus
590:, or
2720:For
2609:Hahn
2465:Maps
2387:Haar
2258:Sub-
2012:Atom
2000:Sets
1878:ISBN
445:for
119:and
34:, a
1811:or
1792:or
1625:if
1455:by
1376:on
1318:on
1180:of
1153:to
633:on
494:or
269:of
46:or
30:In
2815::
1876:,
1815:.
1172:â
1104:,
951:.
940:.
845:.
616:.
564:,
534:,
62:.
42:,
2256:(
2216:(
2181:(
2079:Ï
1989:/
1963:L
1926:e
1919:t
1912:v
1741:0
1738:=
1733:)
1728:)
1725:A
1722:(
1717:1
1710:f
1704:(
1696:=
1693:)
1690:A
1687:(
1675:f
1666:0
1663:=
1660:)
1657:A
1654:(
1645::
1636:A
1613:f
1573:0
1570:=
1567:)
1564:A
1561:(
1550:0
1547:=
1544:)
1541:A
1538:(
1529::
1520:A
1494:,
1481:Ό
1463:f
1421:f
1397:)
1391:,
1388:X
1385:(
1345:.
1337:(
1332:f
1324:f
1320:X
1288:.
1285:X
1279:X
1276::
1270:s
1267:e
1264:m
1261:i
1258:t
1253:n
1243:f
1231:f
1225:f
1218:=
1213:)
1210:n
1207:(
1203:f
1186:n
1182:f
1174:X
1170:X
1166:f
1161:.
1159:R
1155:X
1143:Îł
1135:X
1125:Îł
1114:.
1112:T
1094:T
1090:T
1086:S
1082:T
1075:n
1068:S
1064:λ
1062:(
1060:â
1057:f
1049:λ
1047:(
1045:â
1042:f
1038:λ
1036:(
1034:â
1031:f
1027:S
1023:t
1019:i
1015:t
1013:(
1011:f
1007:S
1003:Ï
999:f
995:Ï
991:λ
987:R
980:λ
976:C
969:S
920:)
917:Y
914:(
911:M
905:)
902:X
899:(
896:M
876:Y
870:X
867::
864:f
833:)
828:2
824:X
820:(
815:1
808:f
804:=
799:1
795:X
771:.
765:d
761:f
755:g
748:1
744:X
735:=
732:)
720:f
716:(
713:d
709:g
702:2
698:X
679:Ό
665:f
659:g
649:Ό
647:(
645:â
642:f
638:2
635:X
631:g
598:f
572:f
543:f
520:1
513:f
469:.
464:2
453:B
432:)
428:)
425:B
422:(
417:1
410:f
405:(
398:=
395:)
392:B
389:(
386:)
380:(
371:f
347:]
341:+
338:,
335:0
332:[
324:2
313:)
307:(
298:f
253:]
247:+
244:,
241:0
238:[
230:1
197:2
193:X
184:1
180:X
173:f
153:)
148:2
140:,
135:2
131:X
127:(
107:)
102:1
94:,
89:1
85:X
81:(
20:)
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