1553:
2721:
1323:
53:, preserving the many advantages of the former in a more general measure-theoretic framework. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of
2291:
579:
1181:
1743:
816:
3073:
1625:
1880:
1548:{\displaystyle {\begin{aligned}{\overline {I}}(h)&=\sup \left\{I(f)\ :\ f\in C,0\leq f\leq h\right\}\\{\overline {\overline {I}}}(h)&=\inf \left\{I(f)\ :\ f\in C,h\leq f\right\}.\end{aligned}}}
2486:
2378:
1328:
288:
226:
2090:
1293:
2950:
2129:
664:
173:
902:
2419:
461:
1017:
3621:
1912:
1649:
1926:. This notion is quite useful for various applications: for example, in muddy terrain the speed in which a person can move may depend on how deep the mud is. If
3516:
3421:
418:
3179:
700:
3499:
3401:
2986:
3251:
1564:
3494:
3416:
3223:
3110:
3396:
2973:
2716:{\displaystyle U(t)V(t)=U(0)V(0)+\int _{(0,t]}U(s-)\,dV(s)+\int _{(0,t]}V(s-)\,dU(s)+\sum _{u\in (0,t]}\Delta U_{u}\Delta V_{u},}
1813:
3411:
3346:
3306:
3147:
3079:
2886:
46:
3406:
3585:
2307:
3570:
3369:
3539:
3506:
243:
181:
2018:
3374:
1223:
2296:
Here the relevant
Lebesgue–Stieltjes measures are associated with the right-continuous versions of the functions
2286:{\displaystyle \int _{a}^{b}U\,dV+\int _{a}^{b}V\,dU=U(b+)V(b+)-U(a-)V(a-),\qquad -\infty <a<b<\infty .}
3384:
1307:
73:
2895:
609:
118:
3244:
1768:
Integrators of bounded variation are handled as above by decomposing into positive and negative variations.
834:
574:{\displaystyle \mu _{g}(E)=\inf \left\{\sum _{i}\mu _{g}(I_{i})\ :\ E\subseteq \bigcup _{i}I_{i}\right\}}
3544:
3446:
3426:
2790:
97:
58:
34:
2383:
3391:
3281:
2120:
452:
50:
3526:
3441:
3436:
3326:
2878:
2465:
1176:{\displaystyle \int _{a}^{b}f(x)\,dg(x)=\int _{a}^{b}f(x)\,dg_{1}(x)-\int _{a}^{b}f(x)\,dg_{2}(x),}
311:
233:
69:
2461:
are of finite variation on this unbounded interval. Complex-valued functions may be used as well.
3549:
3486:
3379:
3351:
3316:
3237:
3127:
2965:
1758:
1317:
on . This functional can then be extended to the class of all non-negative functions by setting
93:
89:
3598:
3575:
3511:
3481:
3473:
3451:
3431:
3321:
3219:
3200:
3175:
2850:
2754:
1888:
1788:
1738:{\displaystyle \mu _{g}(A):={\overline {I}}(\chi _{A})={\overline {\overline {I}}}(\chi _{A})}
670:
307:
291:
54:
17:
3593:
3456:
3341:
3301:
3296:
3291:
3286:
3276:
3119:
2834:
1199:
237:
101:
3580:
3463:
3336:
3171:
1955:
938:
3534:
3205:
2885:
is a non-decreasing real function, the
Lebesgue–Stieltjes integral is equivalent to the
3211:
3159:
31:
3615:
3565:
1963:
1314:
448:
229:
2476:
of finite variation, which are both right-continuous and have left-limits (they are
3311:
3188:
2757:, and is of use in the general theory of stochastic integration. The final term is
85:
3164:
2477:
1630:
and either side of the identity then defines the
Lebesgue–Stieltjes integral of
38:
1186:
where the latter two integrals are well-defined by the preceding construction.
61:, and conversely every regular Borel measure on the real line is of this kind.
592:
by countably many semiopen intervals. This measure is sometimes called the
3108:
Hewitt, Edwin (May 1960). "Integration by Parts for
Stieltjes Integrals".
3260:
65:
3131:
811:{\displaystyle \int _{a}^{b}f(x)\,dg(x):=-\int _{a}^{b}f(x)\,d(-g)(x),}
585:
88:, to whom much of the theory is due. They find common application in
2789:. (The earlier result can then be seen as a result pertaining to the
3123:
3068:{\displaystyle \int _{-\infty }^{\infty }f(x)\,dv(x)=\mathrm {E} .}
2464:
An alternative result, of significant importance in the theory of
1620:{\displaystyle {\overline {I}}(h)={\overline {\overline {I}}}(h),}
1007:
is bounded, the
Lebesgue–Stieltjes integral of f with respect to
821:
the latter integral being defined by the preceding construction.
3233:
1206:
be a non-decreasing right-continuous function on , and define
2103:
of finite variation, if at each point either at least one of
3229:
1875:{\displaystyle \int _{a}^{b}\rho (\gamma (t))\,d\ell (t),}
2955:
for the
Lebesgue–Stieltjes integral, letting the measure
1807:
with respect to the
Euclidean metric weighted by ρ to be
1202:
that extends the usual
Riemann–Stieltjes integral. Let
326:
is monotone non-decreasing and right-continuous. Define
57:
on the real line. The
Lebesgue–Stieltjes measure is a
2310:
1803:
is Borel measurable. Then we may define the length of
1198:) is to define the Lebesgue–Stieltjes integral as the
828:
is of bounded variation, then it is possible to write
3216:
Integral, Measure, and
Derivative: A Unified Approach
2989:
2898:
2489:
2386:
2132:
2021:
1891:
1816:
1652:
1567:
1326:
1226:
1020:
837:
703:
612:
464:
246:
184:
121:
2865:
Riemann–Stieltjes integration and probability theory
1938:
denotes the inverse of the walking speed at or near
3558:
3525:
3472:
3360:
3267:
3218:, Richard A. Silverman, trans. Dover Publications.
2373:{\textstyle {\tilde {U}}(x)=\lim _{t\to x^{+}}U(t)}
2123:formula for the Lebesgue–Stieltjes integral holds:
3163:
3067:
2944:
2715:
2413:
2372:
2285:
2084:
1906:
1874:
1737:
1619:
1547:
1287:
1175:
896:
810:
658:
573:
282:
220:
167:
2964:remain implicit. This is particularly common in
2336:
1962:-length of curves and is useful in the study of
1468:
1357:
487:
2781:which arises from the quadratic covariation of
1195:
283:{\displaystyle g:\left\rightarrow \mathbb {R} }
221:{\displaystyle f:\left\rightarrow \mathbb {R} }
2085:{\displaystyle f(a)={\frac {f(a-)+f(a+)}{2}}.}
3245:
3082:for more detail on dealing with such cases.)
1288:{\displaystyle I(f)=\int _{a}^{b}f(x)\,dg(x)}
8:
2881:real-valued function of a real variable and
2753:. This result can be seen as a precursor to
2433:may be replaced with an unbounded interval
368:(Alternatively, the construction works for
3252:
3238:
3230:
3039:
3020:
3002:
2994:
2988:
2926:
2908:
2903:
2897:
2837:, and the Lebesgue–Stieltjes integral of
2704:
2691:
2660:
2637:
2604:
2581:
2548:
2488:
2388:
2387:
2385:
2350:
2339:
2312:
2311:
2309:
2179:
2170:
2165:
2151:
2142:
2137:
2131:
2037:
2020:
1890:
1853:
1826:
1821:
1815:
1726:
1704:
1692:
1675:
1657:
1651:
1590:
1568:
1566:
1437:
1331:
1327:
1325:
1269:
1251:
1246:
1225:
1155:
1147:
1129:
1124:
1102:
1094:
1076:
1071:
1048:
1030:
1025:
1019:
879:
857:
836:
780:
762:
757:
731:
713:
708:
702:
640:
622:
617:
611:
560:
550:
522:
509:
499:
469:
463:
276:
275:
245:
214:
213:
183:
149:
131:
126:
120:
3517:Common integrals in quantum field theory
2945:{\displaystyle \int _{a}^{b}f(x)\,dv(x)}
1558:For Borel measurable functions, one has
659:{\displaystyle \int _{a}^{b}f(x)\,dg(x)}
168:{\displaystyle \int _{a}^{b}f(x)\,dg(x)}
3622:Definitions of mathematical integration
3427:Differentiation under the integral sign
3091:
2468:is the following. Given two functions
1950:is the time it would take to traverse
897:{\displaystyle g(x)=g_{1}(x)-g_{2}(x)}
1217:to be the Riemann–Stieltjes integral
7:
1914:is the length of the restriction of
3148:Henstock-Kurzweil-Stiltjes Integral
1982:is said to be "regular" at a point
3040:
3003:
2998:
2697:
2684:
2277:
2259:
1986:if the right and left hand limits
1918:to . This is sometimes called the
421:, there is a unique Borel measure
294:in and right-continuous, or when
25:
3111:The American Mathematical Monthly
2976:of a real-valued random variable
2008:exist, and the function takes at
2974:cumulative distribution function
2414:{\displaystyle {\tilde {V}}(x).}
603:The Lebesgue–Stieltjes integral
419:Carathéodory's extension theorem
112:The Lebesgue–Stieltjes integral
2889:, in which case we often write
2255:
694:is non-increasing, then define
3059:
3056:
3050:
3044:
3033:
3027:
3017:
3011:
2939:
2933:
2923:
2917:
2679:
2667:
2650:
2644:
2634:
2625:
2617:
2605:
2594:
2588:
2578:
2569:
2561:
2549:
2538:
2532:
2526:
2520:
2511:
2505:
2499:
2493:
2405:
2399:
2393:
2367:
2361:
2343:
2329:
2323:
2317:
2249:
2240:
2234:
2225:
2216:
2207:
2201:
2192:
2070:
2061:
2052:
2043:
2031:
2025:
1901:
1895:
1866:
1860:
1850:
1847:
1841:
1835:
1732:
1719:
1698:
1685:
1669:
1663:
1611:
1605:
1584:
1578:
1518:
1506:
1485:
1479:
1458:
1452:
1407:
1395:
1374:
1368:
1347:
1341:
1282:
1276:
1266:
1260:
1236:
1230:
1167:
1161:
1144:
1138:
1114:
1108:
1091:
1085:
1061:
1055:
1045:
1039:
891:
885:
869:
863:
847:
841:
802:
796:
793:
784:
777:
771:
744:
738:
728:
722:
653:
647:
637:
631:
528:
515:
481:
475:
272:
210:
162:
156:
146:
140:
43:Lebesgue–Stieltjes integration
27:Lebesgue-Stieltjes integration
18:Lebesgue-Stieltjes integration
1:
3214:, and Gurevich, B. L., 1978.
3080:Riemann–Stieltjes integration
1298:for all continuous functions
996:are monotone non-decreasing.
96:, and in certain branches of
1714:
1710:
1680:
1600:
1596:
1573:
1447:
1443:
1336:
681:with respect to the measure
588:taken over all coverings of
3332:Lebesgue–Stieltjes integral
1196:Hewitt & Stromberg 1965
3638:
3347:Riemann–Stieltjes integral
3307:Henstock–Kurzweil integral
3193:Real and abstract analysis
3191:; Stromberg, Karl (1965),
2887:Riemann–Stieltjes integral
2119:are both regular, then an
594:Lebesgue–Stieltjes measure
3586:Proof that 22/7 exceeds π
1194:An alternative approach (
1958:uses this notion of the
1907:{\displaystyle \ell (t)}
314:. To start, assume that
78:Lebesgue–Radon integrals
74:Thomas Joannes Stieltjes
37:and related branches of
3571:Euler–Maclaurin formula
3206:Theory of the Integral.
178:is defined when
3540:Russo–Vallois integral
3507:Bose–Einstein integral
3422:Parametric derivatives
3098:Halmos (1974), Sec. 15
3069:
2946:
2717:
2415:
2374:
2287:
2086:
1908:
1876:
1739:
1621:
1549:
1289:
1177:
945:in the interval , and
898:
812:
660:
575:
430:on which agrees with
284:
222:
169:
3545:Stratonovich integral
3491:Fermi–Dirac integral
3447:Numerical integration
3070:
2947:
2849:is equivalent to the
2791:Stratonovich integral
2718:
2421:The bounded interval
2416:
2375:
2288:
2087:
1909:
1877:
1740:
1634:. The outer measure
1622:
1550:
1290:
1178:
899:
813:
690:in the usual way. If
661:
576:
285:
223:
170:
59:regular Borel measure
3527:Stochastic integrals
3170:, Berlin, New York:
3078:(See the article on
2987:
2896:
2802:Lebesgue integration
2487:
2384:
2308:
2130:
2121:integration by parts
2095:Given two functions
2019:
1970:Integration by parts
1889:
1814:
1650:
1565:
1324:
1224:
1018:
835:
701:
610:
462:
453:metric outer measure
322:is non-negative and
302:is non-negative and
244:
182:
119:
94:stochastic processes
76:, are also known as
51:Lebesgue integration
3437:Contour integration
3327:Kolmogorov integral
3007:
2913:
2466:stochastic calculus
2175:
2147:
1831:
1256:
1134:
1081:
1035:
767:
718:
627:
136:
70:Henri Leon Lebesgue
64:Lebesgue–Stieltjes
3550:Skorokhod integral
3487:Dirichlet integral
3474:Improper integrals
3417:Reduction formulas
3352:Regulated integral
3317:Hellinger integral
3065:
2990:
2966:probability theory
2942:
2899:
2713:
2683:
2411:
2370:
2357:
2283:
2161:
2133:
2082:
2012:the average value
1964:conformal mappings
1904:
1872:
1817:
1759:indicator function
1735:
1617:
1545:
1543:
1285:
1242:
1173:
1120:
1067:
1021:
894:
808:
753:
704:
669:is defined as the
656:
613:
571:
555:
504:
434:on every interval
280:
218:
165:
122:
3609:
3608:
3512:Frullani integral
3482:Gaussian integral
3432:Laplace transform
3407:Inverse functions
3397:Partial fractions
3322:Khinchin integral
3282:Lebesgue integral
3195:, Springer-Verlag
3181:978-0-387-90088-9
2980:, in which case
2851:Lebesgue integral
2656:
2396:
2335:
2320:
2111:is continuous or
2077:
1954:. The concept of
1791:in the plane and
1789:rectifiable curve
1717:
1713:
1683:
1603:
1599:
1576:
1496:
1490:
1450:
1446:
1385:
1379:
1339:
671:Lebesgue integral
546:
539:
533:
495:
372:left-continuous,
292:bounded variation
55:bounded variation
47:Riemann–Stieltjes
45:generalizes both
32:measure-theoretic
16:(Redirected from
3629:
3457:Trapezoidal rule
3442:Laplace's method
3342:Pfeffer integral
3302:Darboux integral
3297:Daniell integral
3292:Bochner integral
3287:Burkill integral
3277:Riemann integral
3254:
3247:
3240:
3231:
3196:
3184:
3169:
3136:
3135:
3105:
3099:
3096:
3074:
3072:
3071:
3066:
3043:
3006:
3001:
2979:
2971:
2963:
2951:
2949:
2948:
2943:
2912:
2907:
2884:
2876:
2860:
2848:
2845:with respect to
2844:
2835:Lebesgue measure
2832:
2823:
2819:
2797:Related concepts
2788:
2784:
2780:
2752:
2722:
2720:
2719:
2714:
2709:
2708:
2696:
2695:
2682:
2621:
2620:
2565:
2564:
2480:functions) then
2475:
2471:
2460:
2456:
2452:
2448:
2440:
2432:
2420:
2418:
2417:
2412:
2398:
2397:
2389:
2379:
2377:
2376:
2371:
2356:
2355:
2354:
2322:
2321:
2313:
2303:
2299:
2292:
2290:
2289:
2284:
2174:
2169:
2146:
2141:
2118:
2114:
2110:
2106:
2102:
2098:
2091:
2089:
2088:
2083:
2078:
2073:
2038:
2011:
2007:
1996:
1985:
1981:
1961:
1953:
1949:
1945:
1941:
1937:
1925:
1921:
1917:
1913:
1911:
1910:
1905:
1881:
1879:
1878:
1873:
1830:
1825:
1806:
1802:
1786:
1764:
1756:
1744:
1742:
1741:
1736:
1731:
1730:
1718:
1706:
1705:
1697:
1696:
1684:
1676:
1662:
1661:
1642:
1633:
1626:
1624:
1623:
1618:
1604:
1592:
1591:
1577:
1569:
1554:
1552:
1551:
1546:
1544:
1537:
1533:
1494:
1488:
1451:
1439:
1438:
1432:
1428:
1383:
1377:
1340:
1332:
1312:
1305:
1294:
1292:
1291:
1286:
1255:
1250:
1216:
1205:
1200:Daniell integral
1190:Daniell integral
1182:
1180:
1179:
1174:
1160:
1159:
1133:
1128:
1107:
1106:
1080:
1075:
1034:
1029:
1010:
1006:
995:
986:
977:
944:
936:
932:
931:
903:
901:
900:
895:
884:
883:
862:
861:
827:
817:
815:
814:
809:
766:
761:
717:
712:
693:
689:
680:
665:
663:
662:
657:
626:
621:
599:
596:associated with
591:
580:
578:
577:
572:
570:
566:
565:
564:
554:
537:
531:
527:
526:
514:
513:
503:
474:
473:
446:
437:
433:
429:
413:
402:
371:
367:
356:
325:
321:
312:right-continuous
305:
301:
289:
287:
286:
281:
279:
271:
267:
227:
225:
224:
219:
217:
209:
205:
174:
172:
171:
166:
135:
130:
102:potential theory
21:
3637:
3636:
3632:
3631:
3630:
3628:
3627:
3626:
3612:
3611:
3610:
3605:
3581:Integration Bee
3554:
3521:
3468:
3464:Risch algorithm
3402:Euler's formula
3362:
3356:
3337:Pettis integral
3269:
3263:
3258:
3201:Saks, Stanisław
3187:
3182:
3172:Springer-Verlag
3160:Halmos, Paul R.
3158:
3155:
3145:
3140:
3139:
3124:10.2307/2309287
3107:
3106:
3102:
3097:
3093:
3088:
2985:
2984:
2977:
2969:
2961:
2956:
2894:
2893:
2882:
2870:
2867:
2854:
2846:
2838:
2830:
2825:
2821:
2807:
2804:
2799:
2786:
2782:
2758:
2733:
2727:
2700:
2687:
2600:
2544:
2485:
2484:
2473:
2469:
2458:
2454:
2450:
2442:
2434:
2422:
2382:
2381:
2346:
2306:
2305:
2301:
2297:
2128:
2127:
2116:
2112:
2108:
2104:
2100:
2096:
2039:
2017:
2016:
2009:
1998:
1987:
1983:
1975:
1972:
1959:
1956:extremal length
1951:
1947:
1943:
1939:
1927:
1923:
1919:
1915:
1887:
1886:
1812:
1811:
1804:
1792:
1777:
1774:
1762:
1754:
1749:
1722:
1688:
1653:
1648:
1647:
1643:is defined via
1640:
1635:
1631:
1563:
1562:
1542:
1541:
1475:
1471:
1461:
1434:
1433:
1364:
1360:
1350:
1322:
1321:
1310:
1299:
1222:
1221:
1207:
1203:
1192:
1151:
1098:
1016:
1015:
1008:
1000:
994:
988:
985:
979:
963:
952:
946:
942:
939:total variation
930:
925:
924:
923:
914:
908:
875:
853:
833:
832:
825:
699:
698:
691:
687:
682:
674:
608:
607:
597:
589:
556:
518:
505:
494:
490:
465:
460:
459:
447:arises from an
444:
439:
438:. The measure
435:
431:
427:
422:
404:
373:
369:
358:
327:
323:
315:
303:
295:
257:
253:
242:
241:
195:
191:
180:
179:
117:
116:
110:
82:Radon integrals
28:
23:
22:
15:
12:
11:
5:
3635:
3633:
3625:
3624:
3614:
3613:
3607:
3606:
3604:
3603:
3602:
3601:
3596:
3588:
3583:
3578:
3576:Gabriel's horn
3573:
3568:
3562:
3560:
3556:
3555:
3553:
3552:
3547:
3542:
3537:
3531:
3529:
3523:
3522:
3520:
3519:
3514:
3509:
3504:
3503:
3502:
3497:
3489:
3484:
3478:
3476:
3470:
3469:
3467:
3466:
3461:
3460:
3459:
3454:
3452:Simpson's rule
3444:
3439:
3434:
3429:
3424:
3419:
3414:
3412:Changing order
3409:
3404:
3399:
3394:
3389:
3388:
3387:
3382:
3377:
3366:
3364:
3358:
3357:
3355:
3354:
3349:
3344:
3339:
3334:
3329:
3324:
3319:
3314:
3309:
3304:
3299:
3294:
3289:
3284:
3279:
3273:
3271:
3265:
3264:
3259:
3257:
3256:
3249:
3242:
3234:
3228:
3227:
3209:
3198:
3185:
3180:
3166:Measure Theory
3154:
3151:
3144:
3141:
3138:
3137:
3118:(5): 419–423.
3100:
3090:
3089:
3087:
3084:
3076:
3075:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3042:
3038:
3035:
3032:
3029:
3026:
3023:
3019:
3016:
3013:
3010:
3005:
3000:
2997:
2993:
2959:
2953:
2952:
2941:
2938:
2935:
2932:
2929:
2925:
2922:
2919:
2916:
2911:
2906:
2902:
2866:
2863:
2828:
2803:
2800:
2798:
2795:
2731:
2724:
2723:
2712:
2707:
2703:
2699:
2694:
2690:
2686:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2659:
2655:
2652:
2649:
2646:
2643:
2640:
2636:
2633:
2630:
2627:
2624:
2619:
2616:
2613:
2610:
2607:
2603:
2599:
2596:
2593:
2590:
2587:
2584:
2580:
2577:
2574:
2571:
2568:
2563:
2560:
2557:
2554:
2551:
2547:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2453:provided that
2410:
2407:
2404:
2401:
2395:
2392:
2380:and similarly
2369:
2366:
2363:
2360:
2353:
2349:
2345:
2342:
2338:
2334:
2331:
2328:
2325:
2319:
2316:
2304:; that is, to
2294:
2293:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2254:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2230:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2178:
2173:
2168:
2164:
2160:
2157:
2154:
2150:
2145:
2140:
2136:
2093:
2092:
2081:
2076:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2042:
2036:
2033:
2030:
2027:
2024:
1971:
1968:
1903:
1900:
1897:
1894:
1883:
1882:
1871:
1868:
1865:
1862:
1859:
1856:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1829:
1824:
1820:
1773:
1770:
1752:
1746:
1745:
1734:
1729:
1725:
1721:
1716:
1712:
1709:
1703:
1700:
1695:
1691:
1687:
1682:
1679:
1674:
1671:
1668:
1665:
1660:
1656:
1638:
1628:
1627:
1616:
1613:
1610:
1607:
1602:
1598:
1595:
1589:
1586:
1583:
1580:
1575:
1572:
1556:
1555:
1540:
1536:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1493:
1487:
1484:
1481:
1478:
1474:
1470:
1467:
1464:
1462:
1460:
1457:
1454:
1449:
1445:
1442:
1436:
1435:
1431:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1382:
1376:
1373:
1370:
1367:
1363:
1359:
1356:
1353:
1351:
1349:
1346:
1343:
1338:
1335:
1330:
1329:
1296:
1295:
1284:
1281:
1278:
1275:
1272:
1268:
1265:
1262:
1259:
1254:
1249:
1245:
1241:
1238:
1235:
1232:
1229:
1191:
1188:
1184:
1183:
1172:
1169:
1166:
1163:
1158:
1154:
1150:
1146:
1143:
1140:
1137:
1132:
1127:
1123:
1119:
1116:
1113:
1110:
1105:
1101:
1097:
1093:
1090:
1087:
1084:
1079:
1074:
1070:
1066:
1063:
1060:
1057:
1054:
1051:
1047:
1044:
1041:
1038:
1033:
1028:
1024:
1011:is defined by
992:
983:
961:
950:
926:
912:
905:
904:
893:
890:
887:
882:
878:
874:
871:
868:
865:
860:
856:
852:
849:
846:
843:
840:
819:
818:
807:
804:
801:
798:
795:
792:
789:
786:
783:
779:
776:
773:
770:
765:
760:
756:
752:
749:
746:
743:
740:
737:
734:
730:
727:
724:
721:
716:
711:
707:
685:
667:
666:
655:
652:
649:
646:
643:
639:
636:
633:
630:
625:
620:
616:
582:
581:
569:
563:
559:
553:
549:
545:
542:
536:
530:
525:
521:
517:
512:
508:
502:
498:
493:
489:
486:
483:
480:
477:
472:
468:
442:
425:
278:
274:
270:
266:
263:
260:
256:
252:
249:
216:
212:
208:
204:
201:
198:
194:
190:
187:
176:
175:
164:
161:
158:
155:
152:
148:
145:
142:
139:
134:
129:
125:
109:
106:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3634:
3623:
3620:
3619:
3617:
3600:
3597:
3595:
3592:
3591:
3589:
3587:
3584:
3582:
3579:
3577:
3574:
3572:
3569:
3567:
3566:Basel problem
3564:
3563:
3561:
3559:Miscellaneous
3557:
3551:
3548:
3546:
3543:
3541:
3538:
3536:
3533:
3532:
3530:
3528:
3524:
3518:
3515:
3513:
3510:
3508:
3505:
3501:
3498:
3496:
3493:
3492:
3490:
3488:
3485:
3483:
3480:
3479:
3477:
3475:
3471:
3465:
3462:
3458:
3455:
3453:
3450:
3449:
3448:
3445:
3443:
3440:
3438:
3435:
3433:
3430:
3428:
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3386:
3383:
3381:
3378:
3376:
3375:Trigonometric
3373:
3372:
3371:
3368:
3367:
3365:
3359:
3353:
3350:
3348:
3345:
3343:
3340:
3338:
3335:
3333:
3330:
3328:
3325:
3323:
3320:
3318:
3315:
3313:
3312:Haar integral
3310:
3308:
3305:
3303:
3300:
3298:
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3278:
3275:
3274:
3272:
3266:
3262:
3255:
3250:
3248:
3243:
3241:
3236:
3235:
3232:
3225:
3224:0-486-63519-8
3221:
3217:
3213:
3212:Shilov, G. E.
3210:
3208:
3207:
3202:
3199:
3194:
3190:
3189:Hewitt, Edwin
3186:
3183:
3177:
3173:
3168:
3167:
3161:
3157:
3156:
3152:
3150:
3149:
3142:
3133:
3129:
3125:
3121:
3117:
3113:
3112:
3104:
3101:
3095:
3092:
3085:
3083:
3081:
3062:
3053:
3047:
3036:
3030:
3024:
3021:
3014:
3008:
2995:
2991:
2983:
2982:
2981:
2975:
2967:
2962:
2936:
2930:
2927:
2920:
2914:
2909:
2904:
2900:
2892:
2891:
2890:
2888:
2880:
2874:
2864:
2862:
2858:
2852:
2842:
2836:
2831:
2820:for all real
2818:
2814:
2810:
2801:
2796:
2794:
2792:
2778:
2774:
2770:
2766:
2762:
2756:
2750:
2746:
2742:
2738:
2734:
2710:
2705:
2701:
2692:
2688:
2676:
2673:
2670:
2664:
2661:
2657:
2653:
2647:
2641:
2638:
2631:
2628:
2622:
2614:
2611:
2608:
2601:
2597:
2591:
2585:
2582:
2575:
2572:
2566:
2558:
2555:
2552:
2545:
2541:
2535:
2529:
2523:
2517:
2514:
2508:
2502:
2496:
2490:
2483:
2482:
2481:
2479:
2467:
2462:
2446:
2438:
2430:
2426:
2408:
2402:
2390:
2364:
2358:
2351:
2347:
2340:
2332:
2326:
2314:
2280:
2274:
2271:
2268:
2265:
2262:
2256:
2252:
2246:
2243:
2237:
2231:
2228:
2222:
2219:
2213:
2210:
2204:
2198:
2195:
2189:
2186:
2183:
2180:
2176:
2171:
2166:
2162:
2158:
2155:
2152:
2148:
2143:
2138:
2134:
2126:
2125:
2124:
2122:
2079:
2074:
2067:
2064:
2058:
2055:
2049:
2046:
2040:
2034:
2028:
2022:
2015:
2014:
2013:
2005:
2001:
1994:
1990:
1979:
1969:
1967:
1965:
1957:
1935:
1931:
1898:
1892:
1869:
1863:
1857:
1854:
1844:
1838:
1832:
1827:
1822:
1818:
1810:
1809:
1808:
1800:
1796:
1790:
1785:
1781:
1776:Suppose that
1771:
1769:
1766:
1760:
1755:
1727:
1723:
1707:
1701:
1693:
1689:
1677:
1672:
1666:
1658:
1654:
1646:
1645:
1644:
1641:
1614:
1608:
1593:
1587:
1581:
1570:
1561:
1560:
1559:
1538:
1534:
1530:
1527:
1524:
1521:
1515:
1512:
1509:
1503:
1500:
1497:
1491:
1482:
1476:
1472:
1465:
1463:
1455:
1440:
1429:
1425:
1422:
1419:
1416:
1413:
1410:
1404:
1401:
1398:
1392:
1389:
1386:
1380:
1371:
1365:
1361:
1354:
1352:
1344:
1333:
1320:
1319:
1318:
1316:
1315:Radon measure
1309:
1303:
1279:
1273:
1270:
1263:
1257:
1252:
1247:
1243:
1239:
1233:
1227:
1220:
1219:
1218:
1214:
1210:
1201:
1197:
1189:
1187:
1170:
1164:
1156:
1152:
1148:
1141:
1135:
1130:
1125:
1121:
1117:
1111:
1103:
1099:
1095:
1088:
1082:
1077:
1072:
1068:
1064:
1058:
1052:
1049:
1042:
1036:
1031:
1026:
1022:
1014:
1013:
1012:
1004:
997:
991:
982:
975:
971:
967:
960:
956:
949:
940:
935:
929:
922:
918:
911:
888:
880:
876:
872:
866:
858:
854:
850:
844:
838:
831:
830:
829:
822:
805:
799:
790:
787:
781:
774:
768:
763:
758:
754:
750:
747:
741:
735:
732:
725:
719:
714:
709:
705:
697:
696:
695:
688:
678:
672:
650:
644:
641:
634:
628:
623:
618:
614:
606:
605:
604:
601:
595:
587:
567:
561:
557:
551:
547:
543:
540:
534:
523:
519:
510:
506:
500:
496:
491:
484:
478:
470:
466:
458:
457:
456:
454:
450:
449:outer measure
445:
428:
420:
415:
411:
407:
400:
396:
392:
388:
384:
380:
376:
365:
361:
354:
350:
346:
342:
338:
334:
330:
319:
313:
309:
299:
293:
290: is of
268:
264:
261:
258:
254:
250:
247:
239:
235:
231:
206:
202:
199:
196:
192:
188:
185:
159:
153:
150:
143:
137:
132:
127:
123:
115:
114:
113:
107:
105:
103:
99:
95:
91:
87:
83:
79:
75:
71:
67:
62:
60:
56:
52:
48:
44:
40:
36:
33:
19:
3535:Itô integral
3370:Substitution
3361:Integration
3331:
3215:
3204:
3192:
3165:
3146:
3115:
3109:
3103:
3094:
3077:
2957:
2954:
2872:
2868:
2856:
2840:
2826:
2816:
2812:
2808:
2805:
2776:
2772:
2768:
2764:
2760:
2748:
2744:
2740:
2736:
2729:
2725:
2463:
2444:
2436:
2428:
2424:
2295:
2094:
2003:
1999:
1992:
1988:
1977:
1973:
1933:
1929:
1884:
1798:
1794:
1783:
1779:
1775:
1767:
1750:
1747:
1636:
1629:
1557:
1301:
1297:
1212:
1208:
1193:
1185:
1002:
998:
989:
980:
973:
969:
965:
958:
954:
947:
933:
927:
920:
916:
909:
906:
823:
820:
683:
676:
668:
602:
593:
583:
451:(in fact, a
440:
423:
416:
409:
405:
398:
394:
390:
386:
382:
378:
374:
363:
359:
352:
348:
344:
340:
336:
332:
328:
317:
297:
177:
111:
86:Johann Radon
81:
77:
68:, named for
63:
42:
29:
3385:Weierstrass
2755:Itô's lemma
1974:A function
1946:-length of
1942:, then the
1922:-length of
1782: : →
455:) given by
240:and
90:probability
39:mathematics
3500:incomplete
3363:techniques
3153:References
2879:continuous
1313:defines a
1308:functional
234:measurable
228: is
108:Definition
100:including
3270:integrals
3268:Types of
3261:Integrals
3004:∞
2999:∞
2996:−
2992:∫
2901:∫
2698:Δ
2685:Δ
2665:∈
2658:∑
2632:−
2602:∫
2576:−
2546:∫
2394:~
2344:→
2318:~
2278:∞
2260:∞
2257:−
2247:−
2232:−
2220:−
2163:∫
2135:∫
2050:−
1893:ℓ
1858:ℓ
1839:γ
1833:ρ
1819:∫
1724:χ
1715:¯
1711:¯
1690:χ
1681:¯
1655:μ
1601:¯
1597:¯
1574:¯
1528:≤
1501:∈
1448:¯
1444:¯
1423:≤
1417:≤
1390:∈
1337:¯
1244:∫
1122:∫
1118:−
1069:∫
1023:∫
873:−
788:−
755:∫
751:−
706:∫
615:∫
548:⋃
544:⊆
507:μ
497:∑
467:μ
273:→
211:→
124:∫
66:integrals
3616:Category
3590:Volumes
3495:complete
3392:By parts
3162:(1974),
3143:Also see
2002: (
1991: (
1801:→ [0, ∞)
1797: :
1215: )
1211:(
999:Now, if
308:monotone
98:analysis
84:, after
80:or just
35:analysis
3594:Washers
3203:(1937)
3132:2309287
2972:is the
2875:
2871:
2859:
2855:
2843:
2839:
2833:is the
2824:, then
2451:(-∞, ∞)
1980:
1976:
1928:
1793:
1778:
1772:Example
1757:is the
1304:
1300:
1005:
1001:
978:. Both
937:is the
679:
675:
586:infimum
320:
316:
300:
296:
238:bounded
3599:Shells
3222:
3178:
3130:
2869:Where
2726:where
2478:càdlàg
1885:where
1748:where
1495:
1489:
1384:
1378:
1306:. The
907:where
538:
532:
412:}) = 0
366:}) = 0
3380:Euler
3128:JSTOR
3086:Notes
2968:when
2877:is a
2806:When
2435:(-∞,
1787:is a
385:)) =
339:]) =
230:Borel
3220:ISBN
3176:ISBN
2815:) =
2785:and
2775:) =
2743:) −
2472:and
2457:and
2447:, ∞)
2300:and
2275:<
2269:<
2263:<
2115:and
2099:and
1997:and
987:and
968:) −
957:) =
919:) =
584:the
403:and
393:) −
357:and
347:) −
310:and
236:and
92:and
72:and
49:and
3120:doi
2853:of
2793:.)
2449:or
2337:lim
2107:or
1761:of
1469:inf
1358:sup
941:of
824:If
673:of
488:inf
417:By
414:).
306:is
30:In
3618::
3174:,
3126:.
3116:67
3114:.
2861:.
2767:)Δ
2751:−)
2735:=
2441:,
2427:,
2006:−)
1995:+)
1966:.
1765:.
1673::=
748::=
600:.
408:({
377:([
362:({
335:,
331:((
104:.
41:,
3253:e
3246:t
3239:v
3226:.
3197:.
3134:.
3122::
3063:.
3060:]
3057:)
3054:X
3051:(
3048:f
3045:[
3041:E
3037:=
3034:)
3031:x
3028:(
3025:v
3022:d
3018:)
3015:x
3012:(
3009:f
2978:X
2970:v
2960:v
2958:μ
2940:)
2937:x
2934:(
2931:v
2928:d
2924:)
2921:x
2918:(
2915:f
2910:b
2905:a
2883:v
2873:f
2857:f
2847:g
2841:f
2829:g
2827:μ
2822:x
2817:x
2813:x
2811:(
2809:g
2787:V
2783:U
2779:,
2777:d
2773:t
2771:(
2769:V
2765:t
2763:(
2761:U
2759:Δ
2749:t
2747:(
2745:U
2741:t
2739:(
2737:U
2732:t
2730:U
2728:Δ
2711:,
2706:u
2702:V
2693:u
2689:U
2680:]
2677:t
2674:,
2671:0
2668:(
2662:u
2654:+
2651:)
2648:s
2645:(
2642:U
2639:d
2635:)
2629:s
2626:(
2623:V
2618:]
2615:t
2612:,
2609:0
2606:(
2598:+
2595:)
2592:s
2589:(
2586:V
2583:d
2579:)
2573:s
2570:(
2567:U
2562:]
2559:t
2556:,
2553:0
2550:(
2542:+
2539:)
2536:0
2533:(
2530:V
2527:)
2524:0
2521:(
2518:U
2515:=
2512:)
2509:t
2506:(
2503:V
2500:)
2497:t
2494:(
2491:U
2474:V
2470:U
2459:V
2455:U
2445:a
2443:(
2439:)
2437:b
2431:)
2429:b
2425:a
2423:(
2409:.
2406:)
2403:x
2400:(
2391:V
2368:)
2365:t
2362:(
2359:U
2352:+
2348:x
2341:t
2333:=
2330:)
2327:x
2324:(
2315:U
2302:V
2298:U
2281:.
2272:b
2266:a
2253:,
2250:)
2244:a
2241:(
2238:V
2235:)
2229:a
2226:(
2223:U
2217:)
2214:+
2211:b
2208:(
2205:V
2202:)
2199:+
2196:b
2193:(
2190:U
2187:=
2184:U
2181:d
2177:V
2172:b
2167:a
2159:+
2156:V
2153:d
2149:U
2144:b
2139:a
2117:V
2113:U
2109:V
2105:U
2101:V
2097:U
2080:.
2075:2
2071:)
2068:+
2065:a
2062:(
2059:f
2056:+
2053:)
2047:a
2044:(
2041:f
2035:=
2032:)
2029:a
2026:(
2023:f
2010:a
2004:a
2000:f
1993:a
1989:f
1984:a
1978:f
1960:ρ
1952:γ
1948:γ
1944:ρ
1940:z
1936:)
1934:z
1932:(
1930:ρ
1924:γ
1920:ρ
1916:γ
1902:)
1899:t
1896:(
1870:,
1867:)
1864:t
1861:(
1855:d
1851:)
1848:)
1845:t
1842:(
1836:(
1828:b
1823:a
1805:γ
1799:R
1795:ρ
1784:R
1780:γ
1763:A
1753:A
1751:χ
1733:)
1728:A
1720:(
1708:I
1702:=
1699:)
1694:A
1686:(
1678:I
1670:)
1667:A
1664:(
1659:g
1639:g
1637:μ
1632:h
1615:,
1612:)
1609:h
1606:(
1594:I
1588:=
1585:)
1582:h
1579:(
1571:I
1539:.
1535:}
1531:f
1525:h
1522:,
1519:]
1516:b
1513:,
1510:a
1507:[
1504:C
1498:f
1492::
1486:)
1483:f
1480:(
1477:I
1473:{
1466:=
1459:)
1456:h
1453:(
1441:I
1430:}
1426:h
1420:f
1414:0
1411:,
1408:]
1405:b
1402:,
1399:a
1396:[
1393:C
1387:f
1381::
1375:)
1372:f
1369:(
1366:I
1362:{
1355:=
1348:)
1345:h
1342:(
1334:I
1311:I
1302:f
1283:)
1280:x
1277:(
1274:g
1271:d
1267:)
1264:x
1261:(
1258:f
1253:b
1248:a
1240:=
1237:)
1234:f
1231:(
1228:I
1213:f
1209:I
1204:g
1171:,
1168:)
1165:x
1162:(
1157:2
1153:g
1149:d
1145:)
1142:x
1139:(
1136:f
1131:b
1126:a
1115:)
1112:x
1109:(
1104:1
1100:g
1096:d
1092:)
1089:x
1086:(
1083:f
1078:b
1073:a
1065:=
1062:)
1059:x
1056:(
1053:g
1050:d
1046:)
1043:x
1040:(
1037:f
1032:b
1027:a
1009:g
1003:f
993:2
990:g
984:1
981:g
976:)
974:x
972:(
970:g
966:x
964:(
962:1
959:g
955:x
953:(
951:2
948:g
943:g
934:g
928:a
921:V
917:x
915:(
913:1
910:g
892:)
889:x
886:(
881:2
877:g
870:)
867:x
864:(
859:1
855:g
851:=
848:)
845:x
842:(
839:g
826:g
806:,
803:)
800:x
797:(
794:)
791:g
785:(
782:d
778:)
775:x
772:(
769:f
764:b
759:a
745:)
742:x
739:(
736:g
733:d
729:)
726:x
723:(
720:f
715:b
710:a
692:g
686:g
684:μ
677:f
654:)
651:x
648:(
645:g
642:d
638:)
635:x
632:(
629:f
624:b
619:a
598:g
590:E
568:}
562:i
558:I
552:i
541:E
535::
529:)
524:i
520:I
516:(
511:g
501:i
492:{
485:=
482:)
479:E
476:(
471:g
443:g
441:μ
436:I
432:w
426:g
424:μ
410:b
406:w
401:)
399:s
397:(
395:g
391:t
389:(
387:g
383:t
381:,
379:s
375:w
370:g
364:a
360:w
355:)
353:s
351:(
349:g
345:t
343:(
341:g
337:t
333:s
329:w
324:g
318:f
304:g
298:f
277:R
269:]
265:b
262:,
259:a
255:[
251::
248:g
232:-
215:R
207:]
203:b
200:,
197:a
193:[
189::
186:f
163:)
160:x
157:(
154:g
151:d
147:)
144:x
141:(
138:f
133:b
128:a
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.