31:
1105:
As a consequence of the first two properties, it is often possible to reason about "almost every point" of a measure space as though it were an ordinary point rather than an abstraction. This is often done implicitly in informal mathematical arguments. However, one must be careful with this mode of
1708:
in terms of ultrafilters is closely related to the definition in terms of measures, because each ultrafilter defines a finitely-additive measure taking only the values 0 and 1, where a set has measure 1 if and only if it is included in the ultrafilter.
1514:
79:
More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero, or equivalently, if the set of elements for which the property holds is
1394:
1221:
849:
2654:
648:
534:
484:
337:
1441:
200:
1106:
reasoning because of the third bullet above: universal quantification over uncountable families of statements is valid for ordinary points but not for "almost every point".
1099:
930:
882:
794:
3222:
2732:
62:
if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of
270:
3239:
3015:
2749:
1164:
1545:
305:
1013:
1039:
761:
1420:
1285:
1247:
1066:
957:
902:
427:
378:
977:
720:
698:
677:
598:
578:
554:
398:
357:
244:
224:
3039:
2057:
1916:
1816:"On the Convergence Almost Everywhere of Rademacher's Series and of the Bochnerfejér Sums of a Function almost Periodic in the Sense of Stepanoff"
2572:
2403:
1943:
3171:
2822:
2564:
1332:
3234:
2744:
3097:
2350:
1701:
system defines a hyperreal number as an equivalence class of sequences that are equal almost everywhere as defined by an ultrafilter.
3327:
2701:
1885:
1774:
3153:
2691:
1169:
2501:
2174:
3133:
3113:
1611:
Outside of the context of real analysis, the notion of a property true almost everywhere is sometimes defined in terms of an
3229:
2950:
2739:
2030:
3163:
3067:
2686:
2580:
2486:
2605:
2585:
2549:
2473:
2193:
1909:
3217:
2727:
3087:
2506:
2468:
2420:
3181:
2632:
145:
1 not necessarily including all of the outcomes. These are exactly the sets of full measure in a probability space.
3123:
2980:
2600:
2590:
2511:
2478:
2109:
2018:
138:
3186:
2649:
3118:
2554:
2330:
2258:
1590:
As a curiosity, the decimal expansion of almost every real number in the interval contains the complete text of
806:
763:
is a finite or a countable sequence of properties, each of which holds almost everywhere, then their conjunction
3191:
3138:
2639:
148:
Occasionally, instead of saying that a property holds almost everywhere, it is said that the property holds for
3212:
2722:
2168:
2099:
3128:
2035:
3332:
2815:
2491:
2249:
2209:
1902:
3143:
3072:
2965:
2945:
2774:
2674:
2496:
2218:
2064:
1591:
1567:
603:
557:
489:
439:
2970:
2862:
2410:
2335:
2288:
2283:
2278:
2120:
2003:
1961:
797:
724:
55:
1849:
310:
851:
is an uncountable family of properties, each of which holds almost everywhere, then their conjunction
167:
3148:
3034:
2919:
2644:
2610:
2518:
2228:
2183:
2025:
1948:
1071:
907:
854:
766:
3304:
3092:
2975:
2914:
2883:
2627:
2617:
2463:
2427:
2305:
2253:
1982:
1939:
1584:
1326:
87:, it is sufficient that the set be contained within a set of measure zero. When discussing sets of
43:
249:
3274:
3176:
3082:
3029:
2993:
2955:
2888:
2808:
2779:
2539:
2524:
2223:
2104:
2082:
1718:
1299:
556:
that is measurable and has measure zero. However, this technicality vanishes when considering a
73:
1134:
359:. Another common way of expressing the same thing is to say that "almost every point satisfies
1530:
275:
2696:
2432:
2393:
2388:
2295:
2213:
1971:
1881:
1831:
1770:
1128:
982:
1018:
733:
486:
has measure zero; it may not be measurable. By the above definition, it is sufficient that
3264:
3203:
3024:
2924:
2879:
2867:
2713:
2622:
2398:
2383:
2373:
2358:
2325:
2320:
2310:
2188:
2163:
1978:
1823:
1815:
1698:
1580:
1555:. Its complement can be proved to have measure zero. In other words, the Lebesgue mean of
1399:
1255:
1226:
92:
84:
1044:
935:
2789:
2769:
2544:
2442:
2437:
2415:
2273:
2238:
2158:
2052:
887:
403:
111:
1509:{\displaystyle {\frac {1}{2\varepsilon }}\int _{x-\varepsilon }^{x+\varepsilon }f(t)\,dt}
362:
3077:
2893:
2679:
2534:
2529:
2340:
2315:
2268:
2198:
2178:
2138:
2128:
1925:
1250:
962:
705:
683:
662:
583:
563:
539:
383:
342:
229:
209:
51:
39:
1790:
3321:
3289:
3284:
3269:
3259:
2960:
2874:
2849:
2784:
2447:
2368:
2363:
2263:
2233:
2203:
2153:
2148:
2143:
2133:
2047:
1966:
1739:
1599:
203:
68:
3046:
2845:
2378:
2300:
2040:
63:
2077:
1764:
2243:
1612:
142:
88:
1827:
3299:
2990:
2087:
1307:
153:
80:
1835:
125:
is one whose complement is of measure zero. In probability theory, the terms
3294:
3279:
2069:
2013:
2008:
17:
30:
2934:
2903:
2854:
2831:
2094:
1953:
1595:
29:
1894:
2804:
1898:
46:
almost everywhere, more precisely, everywhere except at x = 0.
35:
884:
does not necessarily hold almost everywhere. For example, if
2800:
1598:; similarly for every other finite digit sequence, see
2996:
1850:"Properties That Hold Almost Everywhere - Mathonline"
1533:
1444:
1402:
1335:
1258:
1229:
1172:
1137:
1074:
1047:
1021:
985:
965:
938:
910:
890:
857:
809:
769:
736:
708:
686:
665:
606:
586:
566:
542:
492:
442:
406:
386:
365:
345:
313:
278:
252:
232:
212:
170:
1682:
holds almost everywhere, relative to an ultrafilter
3252:
3200:
3162:
3106:
3055:
2989:
2933:
2902:
2838:
2762:
2710:
2663:
2563:
2456:
2349:
2118:
1991:
1932:
3009:
1791:"Definition of almost everywhere | Dictionary.com"
1721:, a function that is equal to 0 almost everywhere.
1539:
1508:
1414:
1389:{\displaystyle \int _{a}^{b}|f(x)|\,dx<\infty }
1388:
1279:
1241:
1215:
1158:
1093:
1060:
1033:
1007:
971:
951:
924:
896:
876:
843:
788:
755:
714:
692:
671:
642:
592:
572:
548:
528:
478:
421:
392:
372:
351:
331:
299:
264:
238:
218:
194:
1880:(3rd ed.). New York: John Wiley & Sons.
796:holds almost everywhere. This follows from the
723:holds almost everywhere. This follows from the
1820:Proceedings of the London Mathematical Society
2816:
1910:
1068:holds almost everywhere, but the conjunction
679:holds almost everywhere and implies property
8:
2655:RieszâMarkovâKakutani representation theorem
637:
607:
523:
493:
473:
443:
95:is usually assumed unless otherwise stated.
1216:{\displaystyle \int _{a}^{b}f(x)\,dx\geq 0}
844:{\displaystyle (P_{x})_{x\in \mathbf {R} }}
3240:Vitale's random BrunnâMinkowski inequality
3197:
2823:
2809:
2801:
2750:Vitale's random BrunnâMinkowski inequality
2667:
1917:
1903:
1895:
3001:
2995:
1532:
1499:
1475:
1464:
1445:
1443:
1401:
1373:
1368:
1351:
1345:
1340:
1334:
1257:
1228:
1200:
1182:
1177:
1171:
1136:
1085:
1073:
1052:
1046:
1020:
990:
984:
964:
943:
937:
917:
909:
889:
868:
856:
834:
827:
817:
808:
780:
768:
744:
735:
707:
685:
664:
605:
585:
565:
541:
491:
441:
405:
385:
369:
364:
344:
312:
277:
251:
231:
211:
169:
600:exists with measure zero if and only if
1730:
323:
27:Everywhere except a set of measure zero
959:is the property of not being equal to
1697:For example, one construction of the
226:is said to hold almost everywhere in
110:is used, to stand for the equivalent
7:
3253:Applications & related
2763:Applications & related
1657:The intersection of any two sets in
643:{\displaystyle \{x\in X:\neg P(x)\}}
529:{\displaystyle \{x\in X:\neg P(x)\}}
479:{\displaystyle \{x\in X:\neg P(x)\}}
83:. In cases where the measure is not
66:, and is analogous to the notion of
3172:Marcinkiewicz interpolation theorem
3098:Symmetric decreasing rearrangement
3002:
1383:
1075:
858:
770:
622:
508:
458:
259:
180:
25:
1686:, if the set of points for which
650:is measurable with measure zero.
332:{\displaystyle x\in X\setminus N}
246:if there exists a measurable set
2692:Lebesgue differentiation theorem
2573:Carathéodory's extension theorem
918:
835:
195:{\displaystyle (X,\Sigma ,\mu )}
156:can also have other meanings).
1551:is called the Lebesgue set of
1496:
1490:
1369:
1365:
1359:
1352:
1268:
1262:
1197:
1191:
1147:
1141:
1094:{\displaystyle \forall xP_{x}}
1002:
996:
925:{\displaystyle X=\mathbf {R} }
877:{\displaystyle \forall xP_{x}}
824:
810:
789:{\displaystyle \forall nP_{n}}
750:
737:
634:
628:
520:
514:
470:
464:
416:
410:
288:
282:
189:
171:
1:
3068:Convergence almost everywhere
1876:Billingsley, Patrick (1995).
1769:. New York: Springer-Verlag.
1607:Definition using ultrafilters
380:", or that "for almost every
1814:Ursell, H. D. (1932-01-01).
265:{\displaystyle N\in \Sigma }
3235:PrĂ©kopaâLeindler inequality
3088:Locally integrable function
3010:{\displaystyle L^{\infty }}
2745:PrĂ©kopaâLeindler inequality
1547:decreases to zero. The set
3349:
2981:Square-integrable function
2687:Lebesgue's density theorem
1615:. An ultrafilter on a set
1422:, then there exists a set
1159:{\displaystyle f(x)\geq 0}
152:elements (though the term
3230:MinkowskiâSteiner formula
2740:MinkowskiâSteiner formula
2670:
2555:Projection-valued measure
1540:{\displaystyle \epsilon }
536:be contained in some set
300:{\displaystyle \mu (N)=0}
3328:Mathematical terminology
3213:Isoperimetric inequality
2723:Isoperimetric inequality
2702:VitaliâHahnâSaks theorem
2031:Carathéodory's criterion
1828:10.1112/plms/s2-33.1.457
1763:Halmos, Paul R. (1974).
1667:The empty set is not in
1619:is a maximal collection
1166:almost everywhere, then
1008:{\displaystyle P_{x}(y)}
798:countable sub-additivity
3218:BrunnâMinkowski theorem
2728:BrunnâMinkowski theorem
2597:Decomposition theorems
1878:Probability and measure
1101:does not hold anywhere.
1034:{\displaystyle y\neq x}
1015:is true if and only if
904:is Lebesgue measure on
756:{\displaystyle (P_{n})}
3073:Convergence in measure
3011:
2775:Descriptive set theory
2675:Disintegration theorem
2110:Universally measurable
1854:mathonline.wikidot.com
1822:. s2-33 (1): 457â466.
1541:
1510:
1416:
1415:{\displaystyle a<b}
1390:
1281:
1280:{\displaystyle f(x)=0}
1243:
1242:{\displaystyle a<b}
1217:
1160:
1095:
1062:
1035:
1009:
973:
953:
926:
898:
878:
845:
790:
757:
716:
694:
673:
644:
594:
574:
558:complete measure space
550:
530:
480:
436:required that the set
423:
394:
374:
353:
333:
301:
266:
240:
220:
196:
106:; in older literature
47:
3187:RieszâFischer theorem
3012:
2971:Polarization identity
2577:Convergence theorems
2036:Cylindrical Ï-algebra
1744:mathworld.wolfram.com
1583:if and only if it is
1542:
1511:
1417:
1396:for all real numbers
1391:
1282:
1244:
1223:for all real numbers
1218:
1161:
1096:
1063:
1061:{\displaystyle P_{x}}
1036:
1010:
974:
954:
952:{\displaystyle P_{x}}
927:
899:
879:
846:
791:
758:
717:
695:
674:
645:
595:
575:
551:
531:
481:
424:
395:
375:
354:
334:
302:
267:
241:
221:
197:
56:mathematical analysis
33:
3192:RieszâThorin theorem
3035:Infimum and supremum
2994:
2920:Lebesgue integration
2645:Minkowski inequality
2519:Cylinder set measure
2404:Infinite-dimensional
2019:equivalence relation
1949:Lebesgue integration
1719:Dirichlet's function
1531:
1442:
1438:, the Lebesgue mean
1400:
1333:
1256:
1227:
1170:
1135:
1072:
1045:
1019:
983:
963:
936:
908:
897:{\displaystyle \mu }
888:
855:
807:
767:
734:
706:
684:
663:
604:
584:
564:
540:
490:
440:
422:{\displaystyle P(x)}
404:
384:
363:
343:
311:
276:
250:
230:
210:
168:
58:), a property holds
3154:Young's convolution
3093:Measurable function
2976:Pythagorean theorem
2966:Parseval's identity
2915:Integrable function
2640:Hölder's inequality
2502:of random variables
2464:Measurable function
2351:Particular measures
1940:Absolute continuity
1740:"Almost Everywhere"
1738:Weisstein, Eric W.
1592:Shakespeare's plays
1486:
1350:
1327:Lebesgue measurable
1187:
1129:Lebesgue integrable
373:{\displaystyle P\,}
3275:Probability theory
3177:Plancherel theorem
3083:Integral transform
3030:Chebyshev distance
3007:
2956:Euclidean distance
2889:Minkowski distance
2780:Probability theory
2105:Transverse measure
2083:Non-measurable set
2065:Locally measurable
1795:www.dictionary.com
1704:The definition of
1587:almost everywhere.
1581:Riemann integrable
1563:almost everywhere.
1537:
1506:
1460:
1412:
1386:
1336:
1310:almost everywhere.
1300:monotonic function
1287:almost everywhere.
1277:
1239:
1213:
1173:
1156:
1091:
1058:
1031:
1005:
969:
949:
922:
894:
874:
841:
786:
753:
712:
690:
669:
640:
590:
570:
546:
526:
476:
419:
390:
370:
349:
339:have the property
329:
297:
262:
236:
216:
192:
74:probability theory
48:
3315:
3314:
3248:
3247:
3063:Almost everywhere
2848: &
2798:
2797:
2758:
2757:
2487:almost everywhere
2433:Spherical measure
2331:Strictly positive
2259:Projection-valued
1999:Almost everywhere
1972:Probability space
1706:almost everywhere
1458:
972:{\displaystyle x}
715:{\displaystyle Q}
693:{\displaystyle Q}
672:{\displaystyle P}
593:{\displaystyle N}
580:is complete then
573:{\displaystyle X}
549:{\displaystyle N}
393:{\displaystyle x}
352:{\displaystyle P}
239:{\displaystyle X}
219:{\displaystyle P}
100:almost everywhere
60:almost everywhere
16:(Redirected from
3340:
3265:Fourier analysis
3223:Milman's reverse
3206:
3204:Lebesgue measure
3198:
3182:RiemannâLebesgue
3025:Bounded function
3016:
3014:
3013:
3008:
3006:
3005:
2925:Taxicab geometry
2880:Measurable space
2825:
2818:
2811:
2802:
2733:Milman's reverse
2716:
2714:Lebesgue measure
2668:
2072:
2058:infimum/supremum
1979:Measurable space
1919:
1912:
1905:
1896:
1891:
1864:
1863:
1861:
1860:
1846:
1840:
1839:
1811:
1805:
1804:
1802:
1801:
1787:
1781:
1780:
1760:
1754:
1753:
1751:
1750:
1735:
1699:hyperreal number
1578:
1546:
1544:
1543:
1538:
1526:
1515:
1513:
1512:
1507:
1485:
1474:
1459:
1457:
1446:
1430:) such that, if
1421:
1419:
1418:
1413:
1395:
1393:
1392:
1387:
1372:
1355:
1349:
1344:
1286:
1284:
1283:
1278:
1248:
1246:
1245:
1240:
1222:
1220:
1219:
1214:
1186:
1181:
1165:
1163:
1162:
1157:
1100:
1098:
1097:
1092:
1090:
1089:
1067:
1065:
1064:
1059:
1057:
1056:
1040:
1038:
1037:
1032:
1014:
1012:
1011:
1006:
995:
994:
978:
976:
975:
970:
958:
956:
955:
950:
948:
947:
931:
929:
928:
923:
921:
903:
901:
900:
895:
883:
881:
880:
875:
873:
872:
850:
848:
847:
842:
840:
839:
838:
822:
821:
803:By contrast, if
795:
793:
792:
787:
785:
784:
762:
760:
759:
754:
749:
748:
721:
719:
718:
713:
701:, then property
699:
697:
696:
691:
678:
676:
675:
670:
649:
647:
646:
641:
599:
597:
596:
591:
579:
577:
576:
571:
555:
553:
552:
547:
535:
533:
532:
527:
485:
483:
482:
477:
428:
426:
425:
420:
399:
397:
396:
391:
379:
377:
376:
371:
358:
356:
355:
350:
338:
336:
335:
330:
306:
304:
303:
298:
271:
269:
268:
263:
245:
243:
242:
237:
225:
223:
222:
217:
201:
199:
198:
193:
93:Lebesgue measure
21:
3348:
3347:
3343:
3342:
3341:
3339:
3338:
3337:
3318:
3317:
3316:
3311:
3244:
3201:
3196:
3158:
3134:HausdorffâYoung
3114:BabenkoâBeckner
3102:
3051:
2997:
2992:
2991:
2985:
2929:
2898:
2894:Sequence spaces
2834:
2829:
2799:
2794:
2790:Spectral theory
2770:Convex analysis
2754:
2711:
2706:
2659:
2559:
2507:in distribution
2452:
2345:
2175:Logarithmically
2114:
2070:
2053:Essential range
1987:
1928:
1923:
1888:
1875:
1872:
1867:
1858:
1856:
1848:
1847:
1843:
1813:
1812:
1808:
1799:
1797:
1789:
1788:
1784:
1777:
1762:
1761:
1757:
1748:
1746:
1737:
1736:
1732:
1728:
1715:
1609:
1570:
1529:
1528:
1517:
1450:
1440:
1439:
1398:
1397:
1331:
1330:
1254:
1253:
1225:
1224:
1168:
1167:
1133:
1132:
1112:
1081:
1070:
1069:
1048:
1043:
1042:
1017:
1016:
986:
981:
980:
961:
960:
939:
934:
933:
906:
905:
886:
885:
864:
853:
852:
823:
813:
805:
804:
776:
765:
764:
740:
732:
731:
704:
703:
682:
681:
661:
660:
656:
602:
601:
582:
581:
562:
561:
538:
537:
488:
487:
438:
437:
402:
401:
382:
381:
361:
360:
341:
340:
309:
308:
274:
273:
248:
247:
228:
227:
208:
207:
166:
165:
162:
116:presque partout
112:French language
102:is abbreviated
28:
23:
22:
15:
12:
11:
5:
3346:
3344:
3336:
3335:
3333:Measure theory
3330:
3320:
3319:
3313:
3312:
3310:
3309:
3308:
3307:
3302:
3292:
3287:
3282:
3277:
3272:
3267:
3262:
3256:
3254:
3250:
3249:
3246:
3245:
3243:
3242:
3237:
3232:
3227:
3226:
3225:
3215:
3209:
3207:
3195:
3194:
3189:
3184:
3179:
3174:
3168:
3166:
3160:
3159:
3157:
3156:
3151:
3146:
3141:
3136:
3131:
3126:
3121:
3116:
3110:
3108:
3104:
3103:
3101:
3100:
3095:
3090:
3085:
3080:
3078:Function space
3075:
3070:
3065:
3059:
3057:
3053:
3052:
3050:
3049:
3044:
3043:
3042:
3032:
3027:
3021:
3019:
3004:
3000:
2987:
2986:
2984:
2983:
2978:
2973:
2968:
2963:
2958:
2953:
2951:CauchyâSchwarz
2948:
2942:
2940:
2931:
2930:
2928:
2927:
2922:
2917:
2911:
2909:
2900:
2899:
2897:
2896:
2891:
2886:
2877:
2872:
2871:
2870:
2860:
2852:
2850:Hilbert spaces
2842:
2840:
2839:Basic concepts
2836:
2835:
2830:
2828:
2827:
2820:
2813:
2805:
2796:
2795:
2793:
2792:
2787:
2782:
2777:
2772:
2766:
2764:
2760:
2759:
2756:
2755:
2753:
2752:
2747:
2742:
2737:
2736:
2735:
2725:
2719:
2717:
2708:
2707:
2705:
2704:
2699:
2697:Sard's theorem
2694:
2689:
2684:
2683:
2682:
2680:Lifting theory
2671:
2665:
2661:
2660:
2658:
2657:
2652:
2647:
2642:
2637:
2636:
2635:
2633:FubiniâTonelli
2625:
2620:
2615:
2614:
2613:
2608:
2603:
2595:
2594:
2593:
2588:
2583:
2575:
2569:
2567:
2561:
2560:
2558:
2557:
2552:
2547:
2542:
2537:
2532:
2527:
2521:
2516:
2515:
2514:
2512:in probability
2509:
2499:
2494:
2489:
2483:
2482:
2481:
2476:
2471:
2460:
2458:
2454:
2453:
2451:
2450:
2445:
2440:
2435:
2430:
2425:
2424:
2423:
2413:
2408:
2407:
2406:
2396:
2391:
2386:
2381:
2376:
2371:
2366:
2361:
2355:
2353:
2347:
2346:
2344:
2343:
2338:
2333:
2328:
2323:
2318:
2313:
2308:
2303:
2298:
2293:
2292:
2291:
2286:
2281:
2271:
2266:
2261:
2256:
2246:
2241:
2236:
2231:
2226:
2221:
2219:Locally finite
2216:
2206:
2201:
2196:
2191:
2186:
2181:
2171:
2166:
2161:
2156:
2151:
2146:
2141:
2136:
2131:
2125:
2123:
2116:
2115:
2113:
2112:
2107:
2102:
2097:
2092:
2091:
2090:
2080:
2075:
2067:
2062:
2061:
2060:
2050:
2045:
2044:
2043:
2033:
2028:
2023:
2022:
2021:
2011:
2006:
2001:
1995:
1993:
1989:
1988:
1986:
1985:
1976:
1975:
1974:
1964:
1959:
1951:
1946:
1936:
1934:
1933:Basic concepts
1930:
1929:
1926:Measure theory
1924:
1922:
1921:
1914:
1907:
1899:
1893:
1892:
1886:
1871:
1868:
1866:
1865:
1841:
1806:
1782:
1775:
1766:Measure theory
1755:
1729:
1727:
1724:
1723:
1722:
1714:
1711:
1672:
1671:
1665:
1655:
1623:of subsets of
1608:
1605:
1604:
1603:
1588:
1564:
1536:
1505:
1502:
1498:
1495:
1492:
1489:
1484:
1481:
1478:
1473:
1470:
1467:
1463:
1456:
1453:
1449:
1426:(depending on
1411:
1408:
1405:
1385:
1382:
1379:
1376:
1371:
1367:
1364:
1361:
1358:
1354:
1348:
1343:
1339:
1311:
1308:differentiable
1288:
1276:
1273:
1270:
1267:
1264:
1261:
1251:if and only if
1249:with equality
1238:
1235:
1232:
1212:
1209:
1206:
1203:
1199:
1196:
1193:
1190:
1185:
1180:
1176:
1155:
1152:
1149:
1146:
1143:
1140:
1111:
1108:
1103:
1102:
1088:
1084:
1080:
1077:
1055:
1051:
1030:
1027:
1024:
1004:
1001:
998:
993:
989:
968:
946:
942:
920:
916:
913:
893:
871:
867:
863:
860:
837:
833:
830:
826:
820:
816:
812:
801:
783:
779:
775:
772:
752:
747:
743:
739:
728:
711:
689:
668:
655:
652:
639:
636:
633:
630:
627:
624:
621:
618:
615:
612:
609:
589:
569:
545:
525:
522:
519:
516:
513:
510:
507:
504:
501:
498:
495:
475:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
418:
415:
412:
409:
389:
368:
348:
328:
325:
322:
319:
316:
296:
293:
290:
287:
284:
281:
261:
258:
255:
235:
215:
191:
188:
185:
182:
179:
176:
173:
161:
158:
131:almost certain
52:measure theory
40:differentiable
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3345:
3334:
3331:
3329:
3326:
3325:
3323:
3306:
3303:
3301:
3298:
3297:
3296:
3293:
3291:
3290:Sobolev space
3288:
3286:
3285:Real analysis
3283:
3281:
3278:
3276:
3273:
3271:
3270:Lorentz space
3268:
3266:
3263:
3261:
3260:Bochner space
3258:
3257:
3255:
3251:
3241:
3238:
3236:
3233:
3231:
3228:
3224:
3221:
3220:
3219:
3216:
3214:
3211:
3210:
3208:
3205:
3199:
3193:
3190:
3188:
3185:
3183:
3180:
3178:
3175:
3173:
3170:
3169:
3167:
3165:
3161:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3132:
3130:
3127:
3125:
3122:
3120:
3117:
3115:
3112:
3111:
3109:
3105:
3099:
3096:
3094:
3091:
3089:
3086:
3084:
3081:
3079:
3076:
3074:
3071:
3069:
3066:
3064:
3061:
3060:
3058:
3054:
3048:
3045:
3041:
3038:
3037:
3036:
3033:
3031:
3028:
3026:
3023:
3022:
3020:
3018:
2998:
2988:
2982:
2979:
2977:
2974:
2972:
2969:
2967:
2964:
2962:
2961:Hilbert space
2959:
2957:
2954:
2952:
2949:
2947:
2944:
2943:
2941:
2939:
2937:
2932:
2926:
2923:
2921:
2918:
2916:
2913:
2912:
2910:
2908:
2906:
2901:
2895:
2892:
2890:
2887:
2885:
2881:
2878:
2876:
2875:Measure space
2873:
2869:
2866:
2865:
2864:
2861:
2859:
2857:
2853:
2851:
2847:
2844:
2843:
2841:
2837:
2833:
2826:
2821:
2819:
2814:
2812:
2807:
2806:
2803:
2791:
2788:
2786:
2785:Real analysis
2783:
2781:
2778:
2776:
2773:
2771:
2768:
2767:
2765:
2761:
2751:
2748:
2746:
2743:
2741:
2738:
2734:
2731:
2730:
2729:
2726:
2724:
2721:
2720:
2718:
2715:
2709:
2703:
2700:
2698:
2695:
2693:
2690:
2688:
2685:
2681:
2678:
2677:
2676:
2673:
2672:
2669:
2666:
2664:Other results
2662:
2656:
2653:
2651:
2650:RadonâNikodym
2648:
2646:
2643:
2641:
2638:
2634:
2631:
2630:
2629:
2626:
2624:
2623:Fatou's lemma
2621:
2619:
2616:
2612:
2609:
2607:
2604:
2602:
2599:
2598:
2596:
2592:
2589:
2587:
2584:
2582:
2579:
2578:
2576:
2574:
2571:
2570:
2568:
2566:
2562:
2556:
2553:
2551:
2548:
2546:
2543:
2541:
2538:
2536:
2533:
2531:
2528:
2526:
2522:
2520:
2517:
2513:
2510:
2508:
2505:
2504:
2503:
2500:
2498:
2495:
2493:
2490:
2488:
2485:Convergence:
2484:
2480:
2477:
2475:
2472:
2470:
2467:
2466:
2465:
2462:
2461:
2459:
2455:
2449:
2446:
2444:
2441:
2439:
2436:
2434:
2431:
2429:
2426:
2422:
2419:
2418:
2417:
2414:
2412:
2409:
2405:
2402:
2401:
2400:
2397:
2395:
2392:
2390:
2387:
2385:
2382:
2380:
2377:
2375:
2372:
2370:
2367:
2365:
2362:
2360:
2357:
2356:
2354:
2352:
2348:
2342:
2339:
2337:
2334:
2332:
2329:
2327:
2324:
2322:
2319:
2317:
2314:
2312:
2309:
2307:
2304:
2302:
2299:
2297:
2294:
2290:
2289:Outer regular
2287:
2285:
2284:Inner regular
2282:
2280:
2279:Borel regular
2277:
2276:
2275:
2272:
2270:
2267:
2265:
2262:
2260:
2257:
2255:
2251:
2247:
2245:
2242:
2240:
2237:
2235:
2232:
2230:
2227:
2225:
2222:
2220:
2217:
2215:
2211:
2207:
2205:
2202:
2200:
2197:
2195:
2192:
2190:
2187:
2185:
2182:
2180:
2176:
2172:
2170:
2167:
2165:
2162:
2160:
2157:
2155:
2152:
2150:
2147:
2145:
2142:
2140:
2137:
2135:
2132:
2130:
2127:
2126:
2124:
2122:
2117:
2111:
2108:
2106:
2103:
2101:
2098:
2096:
2093:
2089:
2086:
2085:
2084:
2081:
2079:
2076:
2074:
2068:
2066:
2063:
2059:
2056:
2055:
2054:
2051:
2049:
2046:
2042:
2039:
2038:
2037:
2034:
2032:
2029:
2027:
2024:
2020:
2017:
2016:
2015:
2012:
2010:
2007:
2005:
2002:
2000:
1997:
1996:
1994:
1990:
1984:
1980:
1977:
1973:
1970:
1969:
1968:
1967:Measure space
1965:
1963:
1960:
1958:
1956:
1952:
1950:
1947:
1945:
1941:
1938:
1937:
1935:
1931:
1927:
1920:
1915:
1913:
1908:
1906:
1901:
1900:
1897:
1889:
1887:0-471-00710-2
1883:
1879:
1874:
1873:
1869:
1855:
1851:
1845:
1842:
1837:
1833:
1829:
1825:
1821:
1817:
1810:
1807:
1796:
1792:
1786:
1783:
1778:
1776:0-387-90088-8
1772:
1768:
1767:
1759:
1756:
1745:
1741:
1734:
1731:
1725:
1720:
1717:
1716:
1712:
1710:
1707:
1702:
1700:
1695:
1693:
1689:
1685:
1681:
1678:of points in
1677:
1670:
1666:
1664:
1660:
1656:
1654:
1650:
1646:
1642:
1638:
1634:
1630:
1629:
1628:
1626:
1622:
1618:
1614:
1606:
1601:
1600:Normal number
1597:
1594:, encoded in
1593:
1589:
1586:
1582:
1577:
1573:
1569:
1565:
1562:
1559:converges to
1558:
1554:
1550:
1534:
1524:
1520:
1516:converges to
1503:
1500:
1493:
1487:
1482:
1479:
1476:
1471:
1468:
1465:
1461:
1454:
1451:
1447:
1437:
1433:
1429:
1425:
1409:
1406:
1403:
1380:
1377:
1374:
1362:
1356:
1346:
1341:
1337:
1328:
1324:
1320:
1316:
1312:
1309:
1305:
1301:
1297:
1293:
1289:
1274:
1271:
1265:
1259:
1252:
1236:
1233:
1230:
1210:
1207:
1204:
1201:
1194:
1188:
1183:
1178:
1174:
1153:
1150:
1144:
1138:
1131:function and
1130:
1126:
1122:
1118:
1114:
1113:
1109:
1107:
1086:
1082:
1078:
1053:
1049:
1041:), then each
1028:
1025:
1022:
999:
991:
987:
966:
944:
940:
914:
911:
891:
869:
865:
861:
831:
828:
818:
814:
802:
799:
781:
777:
773:
745:
741:
729:
726:
722:
709:
700:
687:
666:
658:
657:
653:
651:
631:
625:
619:
616:
613:
610:
587:
567:
559:
543:
517:
511:
505:
502:
499:
496:
467:
461:
455:
452:
449:
446:
435:
430:
413:
407:
387:
366:
346:
326:
320:
317:
314:
294:
291:
285:
279:
256:
253:
233:
213:
206:, a property
205:
204:measure space
186:
183:
177:
174:
159:
157:
155:
151:
146:
144:
140:
136:
135:almost always
132:
128:
127:almost surely
124:
119:
117:
113:
109:
105:
101:
96:
94:
90:
86:
82:
77:
75:
71:
70:
69:almost surely
65:
61:
57:
54:(a branch of
53:
45:
41:
37:
34:The function
32:
19:
3107:Inequalities
3062:
3047:Uniform norm
2935:
2904:
2855:
2565:Main results
2301:Set function
2229:Metric outer
2184:Decomposable
2041:Cylinder set
1998:
1954:
1877:
1870:Bibliography
1857:. Retrieved
1853:
1844:
1819:
1809:
1798:. Retrieved
1794:
1785:
1765:
1758:
1747:. Retrieved
1743:
1733:
1705:
1703:
1696:
1691:
1690:holds is in
1687:
1683:
1679:
1675:
1673:
1668:
1662:
1658:
1652:
1648:
1644:
1640:
1636:
1632:
1624:
1620:
1616:
1610:
1575:
1571:
1560:
1556:
1552:
1548:
1522:
1518:
1435:
1431:
1427:
1423:
1322:
1318:
1314:
1303:
1295:
1291:
1124:
1120:
1116:
1104:
800:of measures.
727:of measures.
725:monotonicity
702:
680:
659:If property
433:
431:
163:
149:
147:
134:
130:
126:
123:full measure
122:
120:
115:
107:
103:
99:
97:
89:real numbers
78:
67:
64:measure zero
59:
49:
18:Almost every
3305:Von Neumann
3119:Chebyshev's
2525:compact set
2492:of measures
2428:Pushforward
2421:Projections
2411:Logarithmic
2254:Probability
2244:Pre-measure
2026:Borel space
1944:of measures
1674:A property
1627:such that:
1613:ultrafilter
1574: : â
1294: : â
143:probability
121:A set with
3322:Categories
3300:C*-algebra
3124:Clarkson's
2497:in measure
2224:Maximising
2194:Equivalent
2088:Vitali set
1859:2019-11-19
1800:2019-11-19
1749:2019-11-19
1726:References
1585:continuous
1566:A bounded
654:Properties
307:, and all
160:Definition
154:almost all
150:almost all
44:continuous
3295:*-algebra
3280:Quasinorm
3149:Minkowski
3040:Essential
3003:∞
2832:Lp spaces
2611:Maharam's
2581:Dominated
2394:Intensity
2389:Hausdorff
2296:Saturated
2214:Invariant
2119:Types of
2078:Ï-algebra
2048:đ-system
2014:Borel set
2009:Baire set
1836:0024-6115
1535:ϵ
1483:ε
1472:ε
1469:−
1462:∫
1455:ε
1384:∞
1338:∫
1208:≥
1175:∫
1151:≥
1076:∀
1026:≠
892:μ
859:∀
832:∈
771:∀
623:¬
614:∈
509:¬
500:∈
459:¬
450:∈
324:∖
318:∈
280:μ
260:Σ
257:∈
187:μ
181:Σ
137:refer to
98:The term
3144:Markov's
3139:Hölder's
3129:Hanner's
2946:Bessel's
2884:function
2868:Lebesgue
2628:Fubini's
2618:Egorov's
2586:Monotone
2545:variable
2523:Random:
2474:Strongly
2399:Lebesgue
2384:Harmonic
2374:Gaussian
2359:Counting
2326:Spectral
2321:Singular
2311:s-finite
2306:Ï-finite
2189:Discrete
2164:Complete
2121:Measures
2095:Null set
1983:function
1713:See also
1568:function
1317: :
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1110:Examples
429:holds".
85:complete
3164:Results
2863:Measure
2540:process
2535:measure
2530:element
2469:Bochner
2443:Trivial
2438:Tangent
2416:Product
2274:Regular
2252:)
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2177:)
2169:Content
2159:Complex
2100:Support
2073:-system
1962:Measure
1302:, then
114:phrase
3017:spaces
2938:spaces
2907:spaces
2858:spaces
2846:Banach
2606:Jordan
2591:Vitali
2550:vector
2479:Weakly
2341:Vector
2316:Signed
2269:Random
2210:Quasi-
2199:Finite
2179:Convex
2139:Banach
2129:Atomic
1957:spaces
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2204:Inner
2154:Brown
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2144:Besov
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