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