2787:
2575:
2174:
4512:
6377:
4578:
can be used to calculate areas bounded by curves using
Riemann–Stieltjes integrals. The integration strips of Riemann integration are replaced with strips that are non-rectangular in shape. The method is to transform a "Cavaliere region" with a transformation
1011:
2718:
1383:
4135:
1292:
1740:
4498:
1985:
5030:
1862:
2970:
3719:
3529:
3938:
348:
3047:
1532:
525:
202:
3813:
3308:
852:
833:
3220:
2634:
1299:
2021:
for (non-compact) self-adjoint (or more generally, normal) operators in a
Hilbert space. In this theorem, the integral is considered with respect to a spectral family of projections.
1626:
4365:
1179:
5358:
4893:
4570:
4014:
1415:
2569:
1875:
cumulative probability distribution function on the real line, no matter how ill-behaved. In particular, no matter how ill-behaved the cumulative distribution function
1174:
4811:
4767:
4633:
5083:
5058:
4307:
2781:
555:
5190:
5161:
5132:
4726:
4665:
3988:
2817:
2749:
2605:
2535:
2486:
2297:
2268:
2168:
2139:
1140:
1079:
5103:
4831:
4597:
4384:
4256:
4232:
4205:
4181:
4158:
2629:
2506:
2457:
2437:
2417:
2397:
2377:
2357:
2337:
2317:
2239:
2219:
2199:
2110:
1897:
1455:
1435:
727:
695:
667:
647:
575:
375:
228:
128:
76:
4697:
4022:
1111:
627:
260:
108:
5608:
1642:
1996:
1775:
3538:
3351:
6403:
4901:
5362:
1460:
6337:
4262:
or “Devil's staircase”), in either of which cases the
Riemann–Stieltjes integral is not captured by any expression involving derivatives of
2879:
6157:
1749:
does not have a probability density function with respect to
Lebesgue measure. In particular, it does not work if the distribution of
2840:
Riemann–Stieltjes integrals are allowed, then the
Lebesgue integral is not strictly more general than the Riemann–Stieltjes integral.
6192:
3832:
39:. The definition of this integral was first published in 1894 by Stieltjes. It serves as an instructive and useful precursor of the
6040:
5945:
267:
6197:
43:, and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability.
2981:
2241:-direction is pointing upward, then the surface to be considered is like a curved fence. The fence follows the curve traced by
1753:
is discrete (i.e., all of the probability is accounted for by point-masses), and even if the cumulative distribution function
5523:
383:
136:
5950:
5925:
6023:
5775:
3741:
2041:
on , then the integral exists. Because of the integration by part formula, the integral exists also if the condition on
6312:
6302:
6287:
6207:
5855:
354:
5564:
2786:
6347:
6018:
5940:
5717:
5666:
5630:
5553:
5413:
3251:
771:
3161:
6362:
5955:
5920:
2574:
1546:
2571:
correspond to regions of the fence with the greater projection and thereby carry the most weight in the integral.
5935:
3057:
2829:
2068:
is not of bounded variation, then there will be continuous functions which cannot be integrated with respect to
6352:
6327:
6182:
6150:
5830:
5487:
5930:
5495:
5438:
1574:
6292:
6177:
1557:
6109:
2064:
is of bounded variation if and only if it is the difference between two (bounded) monotone functions. If
6317:
6094:
5893:
4372:
2081:
1006:{\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} g(x)=f(b)g(b)-f(a)g(a)-\int _{a}^{b}g(x)\,\mathrm {d} f(x)}
6342:
6322:
6063:
6030:
5898:
4320:
6408:
6381:
6143:
4836:
4518:
2713:{\displaystyle g(x)={\begin{cases}0&{\text{if }}x\leq s\\1&{\text{if }}x>s\\\end{cases}}}
2658:
2173:
1378:{\displaystyle g(x)={\begin{cases}0&{\text{if }}x\leq s\\1&{\text{if }}x>s\\\end{cases}}}
1323:
6332:
6222:
6187:
5908:
5676:
4575:
208:
36:
3997:
2170:
all along orthogonal axes, leads to a geometric interpretation of the
Riemann–Stieltjes integral.
6242:
5768:
3991:
3326:
The
Riemann–Stieltjes integral can be efficiently handled using an appropriate generalization of
1388:
1176:
is
Riemann integrable, then the Riemann–Stieltjes integral is related to the Riemann integral by
5680:
5378:
Bullock, Gregory L. (May 1988). "A Geometric
Interpretation of the Riemann-Stieltjes Integral".
6257:
4772:
4731:
4602:
6282:
6068:
5970:
5407:
4208:
3238:
5706:
4277:
3113:
by the addition of points, rather than from partitions with a finer mesh. Specifically, the
2010:
of continuous functions in an interval as
Riemann–Stieltjes integrals against functions of
5915:
5805:
5696:
5654:
5511:
5503:
5475:
2754:
2540:
2419:-sheet. The Riemann-Stieljes integral is the area of the projection of this fence onto the
1884:
1145:
844:
533:
56:
5166:
5137:
5108:
4702:
4641:
4493:{\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} g(x)=\int _{g(a)}^{g(b)}f(x)\,\mathrm {d} x}
4140:
where the integral on the right-hand side is the standard Riemann integral, assuming that
3964:
2793:
2725:
2581:
2511:
2462:
2273:
2244:
2144:
2115:
1980:{\displaystyle \operatorname {E} \left=\int _{-\infty }^{\infty }x^{n}\,\mathrm {d} g(x).}
1116:
1055:
1027:
8:
6307:
6217:
6202:
6050:
5965:
5960:
5850:
4130:{\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} g(x)=\int _{a}^{b}f(x)g'(x)\,\mathrm {d} x}
3072:
1762:
1287:{\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} g(x)=\int _{a}^{b}f(x)g'(x)\,\mathrm {d} x}
5658:
5063:
5038:
4274:
The standard Riemann integral is a special case of the Riemann–Stieltjes integral where
3067:
extends the Riemann–Stietjes integral to encompass integrands and integrators which are
6227:
6073:
6010:
5903:
5875:
5840:
5761:
5587:
5542:
5463:
5395:
5088:
4816:
4582:
4241:
4217:
4190:
4166:
4143:
3068:
2614:
2491:
2442:
2422:
2402:
2382:
2362:
2342:
2322:
2302:
2224:
2204:
2184:
2095:
1440:
1420:
712:
698:
680:
652:
632:
560:
360:
213:
113:
61:
4670:
2843:
The Riemann–Stieltjes integral also generalizes to the case when either the integrand
1735:{\displaystyle \operatorname {E} =\int _{-\infty }^{\infty }f(x)g'(x)\,\mathrm {d} x.}
1084:
580:
233:
81:
6297:
6237:
6122:
6099:
6035:
6005:
5997:
5975:
5845:
5713:
5662:
5626:
5549:
5533:
5519:
5455:
4769:
exactly once for any shift in the interval. A "Cavaliere region" is then bounded by
4511:
4184:
2837:
2833:
2038:
2011:
702:
40:
5436:
Hildebrandt, T.H. (1938). "Definitions of Stieltjes integrals of the Riemann type".
6232:
6212:
6166:
6117:
5980:
5865:
5825:
5820:
5815:
5810:
5800:
5740:
5579:
5451:
5447:
5391:
5387:
2018:
1857:{\displaystyle \operatorname {E} =\int _{-\infty }^{\infty }f(x)\,\mathrm {d} g(x)}
1561:
32:
28:
5422:
6277:
6252:
6104:
5987:
5860:
5692:
5499:
5471:
4259:
2832:, which generalizes the Riemann–Stieltjes integral in a way analogous to how the
1766:
1550:
1018:
On the other hand, a classical result shows that the integral is well-defined if
706:
6058:
3064:
6267:
6262:
5548:. Translated by Silverman, Richard A. (Revised English ed.). Dover Press.
4368:
1569:
4163:
More generally, the Riemann integral equals the Riemann–Stieltjes integral if
3731:
exists if and only if, for every ε > 0, there exists a partition
709:(however this last is essentially convention). We specifically do not require
6397:
6089:
5459:
4503:
where the integral on the right-hand side is the standard Riemann integral.
6247:
5835:
5537:
5025:{\displaystyle \int _{a(y)}^{b(y)}f(x)\,dx\ =\ \int _{a'}^{b'}f(x)\,dg(x),}
3242:
3053:
2852:
2004:
3083:
A slight generalization is to consider in the above definition partitions
6272:
5606:
Pollard, Henry (1920). "The Stieltjes integral and its generalizations".
5483:
3327:
1015:
and the existence of either integral implies the existence of the other.
729:
to be continuous, which allows for integrals that have point mass terms.
20:
5729:"An inequality of the Hölder type, connected with Stieltjes integration"
5745:
5728:
5591:
5467:
5430:
5399:
5251:
2000:
2017:
The Riemann–Stieltjes integral also appears in the formulation of the
1995:
The Riemann–Stieltjes integral appears in the original formulation of
4238:
everywhere while still being continuous and increasing (for example,
357:(the length of the longest subinterval) of the partitions approaches
5583:
5427:. International series in pure and applied mathematics. McGraw-Hill.
3052:
of the interval . This generalization plays a role in the study of
2965:{\displaystyle \sup \sum _{i}\|g(t_{i-1})-g(t_{i})\|_{X}<\infty }
6135:
5784:
5357:
T. L. Grobler, E. R. Ackermann, A. J. van Zyl & J. C. Olivier
3237:
This generalization exhibits the Riemann–Stieltjes integral as the
52:
2722:
the fence has a rectangular "gate" of width 1 and height equal to
736:(the value of the Riemann–Stieltjes integral) such that for every
3714:{\displaystyle L(P,f,g)=\sum _{i=1}^{n}\,\,\,\inf _{x\in }f(x).}
5339:
3524:{\displaystyle U(P,f,g)=\sum _{i=1}^{n}\,\,\,\sup _{x\in }f(x)}
3933:{\displaystyle \lim _{\operatorname {mesh} (P)\to 0}=0.\quad }
2014:. Later, that theorem was reformulated in terms of measures.
5275:
343:{\displaystyle P=\{a=x_{0}<x_{1}<\cdots <x_{n}=b\}.}
5753:
5214:
1769:
may serve as an example of this failure). But the identity
2706:
1371:
5653:. Translated by Silverman, Richard A. Dover Publications.
5510:
5257:
5202:
4515:
Visualisation of the Cavaliere integral for the function
3248:
A consequence is that with this definition, the integral
3078:
3042:{\displaystyle a=t_{0}\leq t_{1}\leq \cdots \leq t_{n}=b}
2751:. Thus the gate, and its projection, have area equal to
1527:{\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} g(x)=f(s)}
5263:
520:{\displaystyle S(P,f,g)=\sum _{i=0}^{n-1}f(c_{i})\left}
353:
The integral, then, is defined to be the limit, as the
4234:
has jump discontinuities, or may have derivative zero
3254:
197:{\displaystyle \int _{x=a}^{b}f(x)\,\mathrm {d} g(x).}
5651:
Integral, Measure, and Derivative: A unified approach
5609:
The Quarterly Journal of Pure and Applied Mathematics
5169:
5140:
5111:
5091:
5066:
5041:
4904:
4839:
4819:
4775:
4734:
4705:
4673:
4644:
4605:
4585:
4521:
4387:
4323:
4280:
4244:
4220:
4193:
4169:
4160:
can be integrated by the Riemann–Stieltjes integral.
4146:
4025:
4000:
3967:
3835:
3744:
3541:
3354:
3164:
2984:
2882:
2796:
2757:
2728:
2637:
2617:
2584:
2543:
2514:
2494:
2465:
2445:
2425:
2405:
2385:
2365:
2345:
2325:
2305:
2276:
2247:
2227:
2207:
2187:
2147:
2118:
2098:
1990:
1900:
1778:
1645:
1577:
1463:
1443:
1423:
1391:
1302:
1182:
1148:
1119:
1087:
1058:
855:
774:
715:
683:
655:
635:
583:
563:
536:
386:
363:
270:
236:
216:
139:
116:
84:
64:
2975:
the supremum being taken over all finite partitions
2459:
plane — in effect, its "shadow". The slope of
2177:
The basic geometry of the Riemann-Stieljes integral.
1628:
is finite, then the probability density function of
1536:
5708:
A Concise Introduction to the Theory of Integration
5386:(5). Mathematical Association of America: 448–455.
5311:
2072:. In general, the integral is not well-defined if
5705:
5541:
5184:
5155:
5126:
5097:
5077:
5052:
5024:
4887:
4825:
4805:
4761:
4720:
4691:
4659:
4627:
4591:
4564:
4492:
4359:
4301:
4250:
4226:
4199:
4175:
4152:
4129:
4008:
3982:
3932:
3808:{\displaystyle U(P,f,g)-L(P,f,g)<\varepsilon .}
3807:
3713:
3523:
3302:
3214:
3041:
2964:
2811:
2775:
2743:
2712:
2623:
2599:
2563:
2529:
2500:
2488:weights the area of the projection. The values of
2480:
2451:
2431:
2411:
2391:
2371:
2351:
2331:
2311:
2291:
2262:
2233:
2213:
2193:
2162:
2133:
2104:
1979:
1856:
1734:
1620:
1526:
1449:
1429:
1409:
1377:
1286:
1168:
1134:
1105:
1073:
1005:
827:
721:
689:
661:
641:
621:
569:
549:
519:
369:
342:
254:
222:
196:
122:
102:
70:
4378:. Then the Riemann–Stieltjes can be evaluated as
2819:on the geometry of the Riemann-Stieljes integral.
2607:on the geometry of the Riemann-Stieljes integral.
2029:The best simple existence theorem states that if
6395:
5226:
4339:
3837:
3822:is Riemann–Stieltjes integrable with respect to
3649:
3462:
3303:{\textstyle \int _{a}^{b}f(x)\,\mathrm {d} g(x)}
2883:
744: > 0 such that for every partition
5532:
5281:
828:{\displaystyle |S(P,f,g)-A|<\varepsilon .\,}
5648:
5363:Council for Scientific and Industrial Research
3215:{\displaystyle |S(P,f,g)-A|<\varepsilon \,}
732:The "limit" is here understood to be a number
110:with respect to another real-to-real function
6151:
5769:
5644:(Second ed.). New York, NY: McGraw-Hill.
5620:
5482:
5245:
5220:
3943:
2869:, then it is natural to assume that it is of
6193:Grothendieck–Hirzebruch–Riemann–Roch theorem
4354:
4342:
2947:
2896:
2783:the value of the Riemann-Stieljes integral.
2024:
1547:cumulative probability distribution function
334:
277:
5565:"Partial orderings & Moore-Smith limit"
5435:
5329:
4016:it can be shown that there is the equality
3133: > 0 there exists a partition
2087:
6158:
6144:
5776:
5762:
3723:Then the generalized Riemann–Stieltjes of
2270:, and the height of the fence is given by
6338:Riemann–Roch theorem for smooth manifolds
5744:
5675:
5424:The Theory of Functions of Real Variables
5208:
5003:
4950:
4481:
4415:
4118:
4053:
4002:
3919:
3867:
3647:
3643:
3595:
3591:
3590:
3460:
3456:
3408:
3404:
3403:
3318:have a point of discontinuity in common.
3282:
3211:
1956:
1836:
1720:
1612:
1589:
1491:
1275:
1210:
985:
883:
824:
173:
6041:Common integrals in quantum field theory
5681:"Recherches sur les fractions continues"
4510:
4506:
2785:
2573:
2172:
6404:Definitions of mathematical integration
5951:Differentiation under the integral sign
5703:
5649:Shilov, G. E.; Gurevich, B. L. (1978).
5605:
5562:
5514:; Pfaffenberger, William Elmer (2010).
5377:
5317:
5305:
5293:
5258:Johnsonbaugh & Pfaffenberger (2010)
3071:rather than simple functions; see also
1621:{\displaystyle \operatorname {E} \left}
6396:
5420:
5345:
3948:
3115:generalized Riemann–Stieltjes integral
3079:Generalized Riemann–Stieltjes integral
843:The Riemann–Stieltjes integral admits
16:Generalization of the Riemann integral
6139:
5757:
5726:
5639:
5269:
5232:
2836:generalizes the Riemann integral. If
2084:, but there are other cases as well.
6165:
5516:Foundations of mathematical analysis
3338:on define the upper Darboux sum of
3310:can still be defined in cases where
46:
5642:Principles of mathematical analysis
5518:. Mineola, NY: Dover Publications.
5492:Functional analysis and semi-groups
4269:
2828:An important generalization is the
1757:is continuous, it does not work if
78:of a real variable on the interval
13:
6303:Riemannian connection on a surface
6208:Measurable Riemann mapping theorem
4483:
4417:
4120:
4055:
3284:
2959:
2359:axis) that is bounded between the
2299:. The fence is the section of the
1991:Application to functional analysis
1958:
1941:
1936:
1901:
1838:
1819:
1814:
1779:
1745:But this formula does not work if
1722:
1686:
1681:
1646:
1578:
1493:
1277:
1212:
987:
885:
207:Its definition uses a sequence of
175:
14:
6420:
5572:The American Mathematical Monthly
5439:The American Mathematical Monthly
5380:The American Mathematical Monthly
4895:. The area of the region is then
2865:takes values in the Banach space
2830:Lebesgue–Stieltjes integral
2823:
1537:Application to probability theory
756:, and for every choice of points
6376:
6375:
5621:Riesz, F.; Sz. Nagy, B. (1990).
5334:Pollard–Moore–Stieltjes integral
4360:{\displaystyle g(x)=\max\{0,x\}}
4187:of its derivative; in this case
740: > 0, there exists
6288:Riemann's differential equation
6198:Hirzebruch–Riemann–Roch theorem
5351:
5323:
4888:{\displaystyle b(y)=a(y)+(b-a)}
4565:{\displaystyle f(x)=(2x+8)^{3}}
3929:
3321:
6313:Riemann–Hilbert correspondence
6183:Generalized Riemann hypothesis
5452:10.1080/00029890.1938.11990804
5392:10.1080/00029890.1988.11972030
5298:
5287:
5238:
5179:
5173:
5150:
5144:
5121:
5115:
5016:
5010:
5000:
4994:
4947:
4941:
4933:
4927:
4919:
4913:
4882:
4870:
4864:
4858:
4849:
4843:
4800:
4794:
4785:
4779:
4756:
4753:
4747:
4735:
4715:
4709:
4686:
4674:
4654:
4648:
4553:
4537:
4531:
4525:
4478:
4472:
4464:
4458:
4450:
4444:
4430:
4424:
4412:
4406:
4333:
4327:
4290:
4284:
4115:
4109:
4098:
4092:
4068:
4062:
4050:
4044:
3977:
3971:
3920:
3916:
3898:
3889:
3871:
3864:
3856:
3853:
3847:
3793:
3775:
3766:
3748:
3705:
3699:
3691:
3659:
3644:
3640:
3621:
3612:
3599:
3592:
3563:
3545:
3518:
3512:
3504:
3472:
3457:
3453:
3434:
3425:
3412:
3405:
3376:
3358:
3297:
3291:
3279:
3273:
3201:
3191:
3173:
3166:
3142:such that for every partition
2943:
2930:
2921:
2902:
2806:
2800:
2790:The effect of a step function
2767:
2761:
2738:
2732:
2647:
2641:
2594:
2588:
2558:
2552:
2524:
2518:
2475:
2469:
2286:
2280:
2257:
2251:
2157:
2151:
2128:
2122:
1971:
1965:
1891:) exists, then it is equal to
1851:
1845:
1833:
1827:
1800:
1797:
1791:
1785:
1717:
1711:
1700:
1694:
1667:
1664:
1658:
1652:
1604:
1598:
1568:is any function for which the
1521:
1515:
1506:
1500:
1488:
1482:
1312:
1306:
1272:
1266:
1255:
1249:
1225:
1219:
1207:
1201:
1163:
1157:
1129:
1123:
1100:
1088:
1068:
1062:
1000:
994:
982:
976:
952:
946:
940:
934:
925:
919:
913:
907:
898:
892:
880:
874:
811:
801:
783:
776:
616:
584:
509:
496:
487:
468:
457:
444:
408:
390:
249:
237:
188:
182:
170:
164:
97:
85:
1:
6348:Riemann–Siegel theta function
5783:
5496:American Mathematical Society
5412:: CS1 maint: date and year (
5371:
5308:and now standard in analysis.
5282:Kolmogorov & Fomin (1975)
4699:, a "translational function"
3334:and a nondecreasing function
1142:increases monotonically, and
838:
6363:Riemann–von Mangoldt formula
5712:(3rd ed.). Birkhauser.
4312:
4009:{\displaystyle \mathbb {R} }
3826:(in the classical sense) if
2578:The effects of curvature in
2221:plane is horizontal and the
2053:is of bounded variation and
1558:probability density function
669:are respectively called the
7:
5856:Lebesgue–Stieltjes integral
5704:Stroock, Daniel W. (1998).
5246:Riesz & Sz. Nagy (1990)
5221:Hille & Phillips (1974)
3225:for every choice of points
3058:Laplace–Stieltjes transform
1410:{\displaystyle a<s<b}
377:, of the approximating sum
27:is a generalization of the
10:
6425:
6358:Riemann–Stieltjes integral
6353:Riemann–Silberstein vector
6328:Riemann–Liouville integral
5871:Riemann–Stieltjes integral
5831:Henstock–Kurzweil integral
5544:Introductory Real Analysis
3944:Examples and special cases
2871:strongly bounded variation
2049:are inversed, that is, if
25:Riemann–Stieltjes integral
6371:
6293:Riemann's minimal surface
6173:
6110:Proof that 22/7 exceeds π
6082:
6049:
5996:
5884:
5791:
5421:Graves, Lawrence (1946).
4806:{\displaystyle f(x),a(y)}
3245:of partitions of .
2339:curve extended along the
2025:Existence of the integral
6318:Riemann–Hilbert problems
6223:Riemann curvature tensor
6188:Grand Riemann hypothesis
6178:Cauchy–Riemann equations
5512:Johnsonbaugh, Richard F.
5195:
4762:{\displaystyle (x,f(x))}
4628:{\displaystyle g=h^{-1}}
4214:It may be the case that
2088:Geometric interpretation
1038:-Hölder continuous with
37:Thomas Joannes Stieltjes
6243:Riemann mapping theorem
6095:Euler–Maclaurin formula
5685:Ann. Fac. Sci. Toulouse
5563:McShane, E. J. (1952).
2537:has the steepest slope
6343:Riemann–Siegel formula
6323:Riemann–Lebesgue lemma
6258:Riemann series theorem
6064:Russo–Vallois integral
6031:Bose–Einstein integral
5946:Parametric derivatives
5640:Rudin, Walter (1964).
5625:. Dover Publications.
5186:
5157:
5128:
5099:
5079:
5054:
5026:
4889:
4827:
4807:
4763:
4722:
4693:
4661:
4629:
4593:
4572:
4566:
4494:
4361:
4317:Consider the function
4303:
4302:{\displaystyle g(x)=x}
4252:
4228:
4201:
4177:
4154:
4131:
4010:
3990:which is continuously
3984:
3934:
3809:
3715:
3589:
3525:
3402:
3304:
3216:
3043:
2966:
2820:
2813:
2777:
2745:
2714:
2625:
2608:
2601:
2565:
2531:
2502:
2482:
2453:
2433:
2413:
2393:
2373:
2353:
2333:
2313:
2293:
2264:
2235:
2215:
2195:
2178:
2164:
2135:
2106:
1999:which represents the
1981:
1858:
1736:
1622:
1528:
1451:
1431:
1411:
1379:
1288:
1170:
1136:
1107:
1075:
1007:
829:
723:
691:
663:
643:
623:
571:
551:
521:
440:
371:
344:
256:
224:
198:
124:
104:
72:
51:The Riemann–Stieltjes
6283:Riemann zeta function
6069:Stratonovich integral
6015:Fermi–Dirac integral
5971:Numerical integration
5677:Stieltjes, Thomas Jan
5359:Cavaliere integration
5187:
5158:
5129:
5100:
5080:
5055:
5027:
4890:
4828:
4808:
4764:
4723:
4694:
4662:
4638:For a given function
4630:
4594:
4576:Cavalieri's principle
4567:
4514:
4507:Cavalieri integration
4495:
4374:rectified linear unit
4367:used in the study of
4362:
4304:
4253:
4229:
4209:absolutely continuous
4202:
4178:
4155:
4132:
4011:
3985:
3935:
3810:
3716:
3569:
3533:and the lower sum by
3526:
3382:
3305:
3217:
3044:
2967:
2814:
2789:
2778:
2776:{\displaystyle f(s),}
2746:
2715:
2626:
2602:
2577:
2566:
2564:{\displaystyle g'(x)}
2532:
2503:
2483:
2454:
2434:
2414:
2394:
2374:
2354:
2334:
2314:
2294:
2265:
2236:
2216:
2196:
2176:
2165:
2136:
2107:
1982:
1879:of a random variable
1859:
1763:absolutely continuous
1737:
1632:is the derivative of
1623:
1529:
1452:
1432:
1412:
1380:
1289:
1171:
1169:{\displaystyle g'(x)}
1137:
1108:
1076:
1008:
830:
724:
692:
664:
644:
629:. The two functions
624:
572:
552:
550:{\displaystyle c_{i}}
522:
414:
372:
345:
257:
225:
199:
125:
105:
73:
6333:Riemann–Roch theorem
6051:Stochastic integrals
5727:Young, L.C. (1936).
5185:{\displaystyle f(x)}
5167:
5156:{\displaystyle b(y)}
5138:
5127:{\displaystyle a(y)}
5109:
5089:
5064:
5039:
4902:
4837:
4817:
4773:
4732:
4721:{\displaystyle a(y)}
4703:
4671:
4660:{\displaystyle f(x)}
4642:
4603:
4583:
4519:
4385:
4321:
4278:
4242:
4218:
4191:
4167:
4144:
4023:
3998:
3983:{\displaystyle g(x)}
3965:
3833:
3742:
3539:
3352:
3252:
3162:
3129:such that for every
3069:stochastic processes
2982:
2880:
2812:{\displaystyle g(x)}
2794:
2755:
2744:{\displaystyle f(s)}
2726:
2635:
2615:
2600:{\displaystyle g(x)}
2582:
2541:
2530:{\displaystyle g(x)}
2512:
2492:
2481:{\displaystyle g(x)}
2463:
2443:
2423:
2403:
2383:
2363:
2343:
2323:
2303:
2292:{\displaystyle f(x)}
2274:
2263:{\displaystyle g(x)}
2245:
2225:
2205:
2185:
2163:{\displaystyle g(x)}
2145:
2134:{\displaystyle f(x)}
2116:
2096:
2080:share any points of
1898:
1776:
1643:
1575:
1461:
1441:
1421:
1389:
1300:
1296:For a step function
1180:
1146:
1135:{\displaystyle g(x)}
1117:
1085:
1074:{\displaystyle f(x)}
1056:
853:
845:integration by parts
772:
713:
707:right-semicontinuous
681:
653:
633:
581:
561:
534:
384:
361:
268:
234:
214:
137:
114:
82:
62:
57:real-valued function
6308:Riemannian geometry
6218:Riemann Xi function
6203:Local zeta function
5961:Contour integration
5851:Kolmogorov integral
5659:1966imdu.book.....S
5623:Functional Analysis
5272:, pp. 121–122.
4990:
4937:
4468:
4402:
4088:
4040:
3330:. For a partition
3269:
3073:stochastic calculus
2631:is a step function
1945:
1823:
1690:
1478:
1245:
1197:
972:
870:
160:
6228:Riemann hypothesis
6074:Skorokhod integral
6011:Dirichlet integral
5998:Improper integrals
5941:Reduction formulas
5876:Regulated integral
5841:Hellinger integral
5746:10.1007/bf02401743
5534:Kolmogorov, Andrey
5494:. Providence, RI:
5488:Phillips, Ralph S.
5330:Hildebrandt (1938)
5182:
5153:
5124:
5095:
5078:{\displaystyle b'}
5075:
5053:{\displaystyle a'}
5050:
5022:
4966:
4905:
4885:
4823:
4803:
4759:
4718:
4689:
4657:
4625:
4589:
4573:
4562:
4490:
4436:
4388:
4357:
4299:
4248:
4224:
4197:
4173:
4150:
4127:
4074:
4026:
4006:
3980:
3930:
3863:
3805:
3711:
3695:
3521:
3508:
3300:
3255:
3212:
3091:another partition
3039:
2962:
2895:
2847:or the integrator
2821:
2809:
2773:
2741:
2710:
2705:
2621:
2609:
2597:
2561:
2527:
2498:
2478:
2449:
2429:
2409:
2389:
2369:
2349:
2329:
2319:-sheet (i.e., the
2309:
2289:
2260:
2231:
2211:
2191:
2179:
2160:
2131:
2102:
2033:is continuous and
1997:F. Riesz's theorem
1977:
1928:
1854:
1806:
1732:
1673:
1618:
1524:
1464:
1447:
1427:
1407:
1375:
1370:
1284:
1231:
1183:
1166:
1132:
1103:
1071:
1003:
958:
856:
825:
719:
687:
659:
639:
619:
567:
547:
517:
367:
340:
252:
220:
194:
140:
120:
100:
68:
6391:
6390:
6298:Riemannian circle
6238:Riemann invariant
6133:
6132:
6036:Frullani integral
6006:Gaussian integral
5956:Laplace transform
5931:Inverse functions
5921:Partial fractions
5846:Khinchin integral
5806:Lebesgue integral
5525:978-0-486-47766-4
5211:, pp. 68–71.
5098:{\displaystyle x}
4965:
4959:
4826:{\displaystyle x}
4592:{\displaystyle h}
4251:{\displaystyle g}
4227:{\displaystyle g}
4200:{\displaystyle g}
4185:Lebesgue integral
4176:{\displaystyle g}
4153:{\displaystyle f}
3836:
3648:
3461:
3239:Moore–Smith limit
2886:
2860: : →
2851:take values in a
2834:Lebesgue integral
2692:
2669:
2624:{\displaystyle g}
2501:{\displaystyle x}
2452:{\displaystyle g}
2432:{\displaystyle f}
2412:{\displaystyle f}
2392:{\displaystyle x}
2372:{\displaystyle g}
2352:{\displaystyle f}
2332:{\displaystyle g}
2312:{\displaystyle g}
2234:{\displaystyle f}
2214:{\displaystyle x}
2194:{\displaystyle g}
2105:{\displaystyle x}
2039:bounded variation
2012:bounded variation
1450:{\displaystyle s}
1437:is continuous at
1430:{\displaystyle f}
1357:
1334:
1028:Hölder continuous
752:) <
722:{\displaystyle g}
703:bounded variation
690:{\displaystyle g}
662:{\displaystyle g}
642:{\displaystyle f}
570:{\displaystyle i}
370:{\displaystyle 0}
223:{\displaystyle P}
123:{\displaystyle g}
71:{\displaystyle f}
47:Formal definition
41:Lebesgue integral
6416:
6409:Bernhard Riemann
6379:
6378:
6233:Riemann integral
6213:Riemann (crater)
6167:Bernhard Riemann
6160:
6153:
6146:
6137:
6136:
5981:Trapezoidal rule
5966:Laplace's method
5866:Pfeffer integral
5826:Darboux integral
5821:Daniell integral
5816:Bochner integral
5811:Burkill integral
5801:Riemann integral
5778:
5771:
5764:
5755:
5754:
5750:
5748:
5733:Acta Mathematica
5723:
5711:
5700:
5672:
5645:
5636:
5617:
5602:
5600:
5598:
5569:
5559:
5547:
5538:Fomin, Sergei V.
5529:
5507:
5479:
5428:
5417:
5411:
5403:
5365:
5355:
5349:
5348:, Chap. XII, §3.
5343:
5337:
5327:
5321:
5315:
5309:
5302:
5296:
5291:
5285:
5279:
5273:
5267:
5261:
5255:
5249:
5242:
5236:
5230:
5224:
5218:
5212:
5209:Stieltjes (1894)
5206:
5191:
5189:
5188:
5183:
5162:
5160:
5159:
5154:
5133:
5131:
5130:
5125:
5104:
5102:
5101:
5096:
5084:
5082:
5081:
5076:
5074:
5059:
5057:
5056:
5051:
5049:
5031:
5029:
5028:
5023:
4989:
4988:
4979:
4978:
4963:
4957:
4936:
4922:
4894:
4892:
4891:
4886:
4832:
4830:
4829:
4824:
4812:
4810:
4809:
4804:
4768:
4766:
4765:
4760:
4727:
4725:
4724:
4719:
4698:
4696:
4695:
4692:{\displaystyle }
4690:
4666:
4664:
4663:
4658:
4634:
4632:
4631:
4626:
4624:
4623:
4598:
4596:
4595:
4590:
4571:
4569:
4568:
4563:
4561:
4560:
4499:
4497:
4496:
4491:
4486:
4467:
4453:
4420:
4401:
4396:
4366:
4364:
4363:
4358:
4308:
4306:
4305:
4300:
4270:Riemann integral
4257:
4255:
4254:
4249:
4233:
4231:
4230:
4225:
4206:
4204:
4203:
4198:
4182:
4180:
4179:
4174:
4159:
4157:
4156:
4151:
4136:
4134:
4133:
4128:
4123:
4108:
4087:
4082:
4058:
4039:
4034:
4015:
4013:
4012:
4007:
4005:
3989:
3987:
3986:
3981:
3939:
3937:
3936:
3931:
3862:
3814:
3812:
3811:
3806:
3727:with respect to
3720:
3718:
3717:
3712:
3694:
3690:
3689:
3677:
3676:
3639:
3638:
3611:
3610:
3588:
3583:
3530:
3528:
3527:
3522:
3507:
3503:
3502:
3490:
3489:
3452:
3451:
3424:
3423:
3401:
3396:
3342:with respect to
3309:
3307:
3306:
3301:
3287:
3268:
3263:
3221:
3219:
3218:
3213:
3204:
3169:
3121:with respect to
3048:
3046:
3045:
3040:
3032:
3031:
3013:
3012:
3000:
2999:
2971:
2969:
2968:
2963:
2955:
2954:
2942:
2941:
2920:
2919:
2894:
2864:
2818:
2816:
2815:
2810:
2782:
2780:
2779:
2774:
2750:
2748:
2747:
2742:
2719:
2717:
2716:
2711:
2709:
2708:
2693:
2690:
2670:
2667:
2630:
2628:
2627:
2622:
2606:
2604:
2603:
2598:
2570:
2568:
2567:
2562:
2551:
2536:
2534:
2533:
2528:
2507:
2505:
2504:
2499:
2487:
2485:
2484:
2479:
2458:
2456:
2455:
2450:
2438:
2436:
2435:
2430:
2418:
2416:
2415:
2410:
2398:
2396:
2395:
2390:
2378:
2376:
2375:
2370:
2358:
2356:
2355:
2350:
2338:
2336:
2335:
2330:
2318:
2316:
2315:
2310:
2298:
2296:
2295:
2290:
2269:
2267:
2266:
2261:
2240:
2238:
2237:
2232:
2220:
2218:
2217:
2212:
2200:
2198:
2197:
2192:
2169:
2167:
2166:
2161:
2140:
2138:
2137:
2132:
2111:
2109:
2108:
2103:
2092:A 3D plot, with
2057:is continuous.
2019:spectral theorem
1986:
1984:
1983:
1978:
1961:
1955:
1954:
1944:
1939:
1924:
1920:
1919:
1863:
1861:
1860:
1855:
1841:
1822:
1817:
1741:
1739:
1738:
1733:
1725:
1710:
1689:
1684:
1627:
1625:
1624:
1619:
1617:
1613:
1611:
1607:
1562:Lebesgue measure
1560:with respect to
1533:
1531:
1530:
1525:
1496:
1477:
1472:
1456:
1454:
1453:
1448:
1436:
1434:
1433:
1428:
1416:
1414:
1413:
1408:
1384:
1382:
1381:
1376:
1374:
1373:
1358:
1355:
1335:
1332:
1293:
1291:
1290:
1285:
1280:
1265:
1244:
1239:
1215:
1196:
1191:
1175:
1173:
1172:
1167:
1156:
1141:
1139:
1138:
1133:
1112:
1110:
1109:
1106:{\displaystyle }
1104:
1080:
1078:
1077:
1072:
1048:
1012:
1010:
1009:
1004:
990:
971:
966:
888:
869:
864:
834:
832:
831:
826:
814:
779:
728:
726:
725:
720:
701:(or at least of
696:
694:
693:
688:
668:
666:
665:
660:
648:
646:
645:
640:
628:
626:
625:
622:{\displaystyle }
620:
615:
614:
596:
595:
577:-th subinterval
576:
574:
573:
568:
556:
554:
553:
548:
546:
545:
526:
524:
523:
518:
516:
512:
508:
507:
486:
485:
456:
455:
439:
428:
376:
374:
373:
368:
349:
347:
346:
341:
327:
326:
308:
307:
295:
294:
261:
259:
258:
255:{\displaystyle }
253:
230:of the interval
229:
227:
226:
221:
203:
201:
200:
195:
178:
159:
154:
129:
127:
126:
121:
109:
107:
106:
103:{\displaystyle }
101:
77:
75:
74:
69:
33:Bernhard Riemann
29:Riemann integral
6424:
6423:
6419:
6418:
6417:
6415:
6414:
6413:
6394:
6393:
6392:
6387:
6367:
6278:Riemann surface
6253:Riemann problem
6169:
6164:
6134:
6129:
6105:Integration Bee
6078:
6045:
5992:
5988:Risch algorithm
5926:Euler's formula
5886:
5880:
5861:Pettis integral
5793:
5787:
5782:
5720:
5669:
5633:
5596:
5594:
5584:10.2307/2307181
5567:
5556:
5526:
5405:
5404:
5374:
5369:
5368:
5356:
5352:
5344:
5340:
5328:
5324:
5316:
5312:
5303:
5299:
5292:
5288:
5280:
5276:
5268:
5264:
5256:
5252:
5243:
5239:
5231:
5227:
5219:
5215:
5207:
5203:
5198:
5168:
5165:
5164:
5139:
5136:
5135:
5110:
5107:
5106:
5105:-values where
5090:
5087:
5086:
5067:
5065:
5062:
5061:
5042:
5040:
5037:
5036:
4981:
4980:
4971:
4970:
4923:
4909:
4903:
4900:
4899:
4838:
4835:
4834:
4818:
4815:
4814:
4774:
4771:
4770:
4733:
4730:
4729:
4728:must intersect
4704:
4701:
4700:
4672:
4669:
4668:
4667:on an interval
4643:
4640:
4639:
4616:
4612:
4604:
4601:
4600:
4584:
4581:
4580:
4556:
4552:
4520:
4517:
4516:
4509:
4482:
4454:
4440:
4416:
4397:
4392:
4386:
4383:
4382:
4369:neural networks
4322:
4319:
4318:
4315:
4279:
4276:
4275:
4272:
4260:Cantor function
4243:
4240:
4239:
4219:
4216:
4215:
4192:
4189:
4188:
4168:
4165:
4164:
4145:
4142:
4141:
4119:
4101:
4083:
4078:
4054:
4035:
4030:
4024:
4021:
4020:
4001:
3999:
3996:
3995:
3966:
3963:
3962:
3959:
3949:Differentiable
3946:
3840:
3834:
3831:
3830:
3743:
3740:
3739:
3685:
3681:
3666:
3662:
3652:
3628:
3624:
3606:
3602:
3584:
3573:
3540:
3537:
3536:
3498:
3494:
3479:
3475:
3465:
3441:
3437:
3419:
3415:
3397:
3386:
3353:
3350:
3349:
3324:
3283:
3264:
3259:
3253:
3250:
3249:
3233:
3200:
3165:
3163:
3160:
3159:
3154:
3141:
3112:
3100:, meaning that
3099:
3081:
3027:
3023:
3008:
3004:
2995:
2991:
2983:
2980:
2979:
2950:
2946:
2937:
2933:
2909:
2905:
2890:
2881:
2878:
2877:
2873:, meaning that
2856:
2826:
2795:
2792:
2791:
2756:
2753:
2752:
2727:
2724:
2723:
2704:
2703:
2689:
2687:
2681:
2680:
2666:
2664:
2654:
2653:
2636:
2633:
2632:
2616:
2613:
2612:
2583:
2580:
2579:
2544:
2542:
2539:
2538:
2513:
2510:
2509:
2493:
2490:
2489:
2464:
2461:
2460:
2444:
2441:
2440:
2424:
2421:
2420:
2404:
2401:
2400:
2384:
2381:
2380:
2364:
2361:
2360:
2344:
2341:
2340:
2324:
2321:
2320:
2304:
2301:
2300:
2275:
2272:
2271:
2246:
2243:
2242:
2226:
2223:
2222:
2206:
2203:
2202:
2186:
2183:
2182:
2146:
2143:
2142:
2117:
2114:
2113:
2097:
2094:
2093:
2090:
2027:
1993:
1957:
1950:
1946:
1940:
1932:
1915:
1911:
1907:
1899:
1896:
1895:
1837:
1818:
1810:
1777:
1774:
1773:
1767:Cantor function
1721:
1703:
1685:
1677:
1644:
1641:
1640:
1594:
1590:
1588:
1584:
1576:
1573:
1572:
1551:random variable
1539:
1492:
1473:
1468:
1462:
1459:
1458:
1442:
1439:
1438:
1422:
1419:
1418:
1390:
1387:
1386:
1369:
1368:
1354:
1352:
1346:
1345:
1331:
1329:
1319:
1318:
1301:
1298:
1297:
1276:
1258:
1240:
1235:
1211:
1192:
1187:
1181:
1178:
1177:
1149:
1147:
1144:
1143:
1118:
1115:
1114:
1086:
1083:
1082:
1057:
1054:
1053:
1039:
986:
967:
962:
884:
865:
860:
854:
851:
850:
841:
810:
775:
773:
770:
769:
764:
714:
711:
710:
697:is taken to be
682:
679:
678:
654:
651:
650:
634:
631:
630:
604:
600:
591:
587:
582:
579:
578:
562:
559:
558:
541:
537:
535:
532:
531:
503:
499:
475:
471:
464:
460:
451:
447:
429:
418:
385:
382:
381:
362:
359:
358:
322:
318:
303:
299:
290:
286:
269:
266:
265:
235:
232:
231:
215:
212:
211:
174:
155:
144:
138:
135:
134:
115:
112:
111:
83:
80:
79:
63:
60:
59:
49:
17:
12:
11:
5:
6422:
6412:
6411:
6406:
6389:
6388:
6386:
6385:
6372:
6369:
6368:
6366:
6365:
6360:
6355:
6350:
6345:
6340:
6335:
6330:
6325:
6320:
6315:
6310:
6305:
6300:
6295:
6290:
6285:
6280:
6275:
6270:
6268:Riemann sphere
6265:
6263:Riemann solver
6260:
6255:
6250:
6245:
6240:
6235:
6230:
6225:
6220:
6215:
6210:
6205:
6200:
6195:
6190:
6185:
6180:
6174:
6171:
6170:
6163:
6162:
6155:
6148:
6140:
6131:
6130:
6128:
6127:
6126:
6125:
6120:
6112:
6107:
6102:
6100:Gabriel's horn
6097:
6092:
6086:
6084:
6080:
6079:
6077:
6076:
6071:
6066:
6061:
6055:
6053:
6047:
6046:
6044:
6043:
6038:
6033:
6028:
6027:
6026:
6021:
6013:
6008:
6002:
6000:
5994:
5993:
5991:
5990:
5985:
5984:
5983:
5978:
5976:Simpson's rule
5968:
5963:
5958:
5953:
5948:
5943:
5938:
5936:Changing order
5933:
5928:
5923:
5918:
5913:
5912:
5911:
5906:
5901:
5890:
5888:
5882:
5881:
5879:
5878:
5873:
5868:
5863:
5858:
5853:
5848:
5843:
5838:
5833:
5828:
5823:
5818:
5813:
5808:
5803:
5797:
5795:
5789:
5788:
5781:
5780:
5773:
5766:
5758:
5752:
5751:
5739:(1): 251–282.
5724:
5718:
5701:
5673:
5667:
5646:
5637:
5631:
5618:
5603:
5560:
5554:
5530:
5524:
5508:
5480:
5446:(5): 265–278.
5433:
5418:
5373:
5370:
5367:
5366:
5350:
5338:
5332:calls it the
5322:
5318:McShane (1952)
5310:
5306:Pollard (1920)
5304:Introduced by
5297:
5294:Bullock (1988)
5286:
5284:, p. 368.
5274:
5262:
5260:, p. 219.
5250:
5237:
5225:
5213:
5200:
5199:
5197:
5194:
5181:
5178:
5175:
5172:
5152:
5149:
5146:
5143:
5123:
5120:
5117:
5114:
5094:
5073:
5070:
5048:
5045:
5033:
5032:
5021:
5018:
5015:
5012:
5009:
5006:
5002:
4999:
4996:
4993:
4987:
4984:
4977:
4974:
4969:
4962:
4956:
4953:
4949:
4946:
4943:
4940:
4935:
4932:
4929:
4926:
4921:
4918:
4915:
4912:
4908:
4884:
4881:
4878:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4822:
4802:
4799:
4796:
4793:
4790:
4787:
4784:
4781:
4778:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4717:
4714:
4711:
4708:
4688:
4685:
4682:
4679:
4676:
4656:
4653:
4650:
4647:
4635:as integrand.
4622:
4619:
4615:
4611:
4608:
4588:
4559:
4555:
4551:
4548:
4545:
4542:
4539:
4536:
4533:
4530:
4527:
4524:
4508:
4505:
4501:
4500:
4489:
4485:
4480:
4477:
4474:
4471:
4466:
4463:
4460:
4457:
4452:
4449:
4446:
4443:
4439:
4435:
4432:
4429:
4426:
4423:
4419:
4414:
4411:
4408:
4405:
4400:
4395:
4391:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4335:
4332:
4329:
4326:
4314:
4311:
4298:
4295:
4292:
4289:
4286:
4283:
4271:
4268:
4247:
4223:
4207:is said to be
4196:
4172:
4149:
4138:
4137:
4126:
4122:
4117:
4114:
4111:
4107:
4104:
4100:
4097:
4094:
4091:
4086:
4081:
4077:
4073:
4070:
4067:
4064:
4061:
4057:
4052:
4049:
4046:
4043:
4038:
4033:
4029:
4004:
3992:differentiable
3979:
3976:
3973:
3970:
3958:
3947:
3945:
3942:
3941:
3940:
3928:
3925:
3922:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3879:
3876:
3873:
3870:
3866:
3861:
3858:
3855:
3852:
3849:
3846:
3843:
3839:
3816:
3815:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3710:
3707:
3704:
3701:
3698:
3693:
3688:
3684:
3680:
3675:
3672:
3669:
3665:
3661:
3658:
3655:
3651:
3646:
3642:
3637:
3634:
3631:
3627:
3623:
3620:
3617:
3614:
3609:
3605:
3601:
3598:
3594:
3587:
3582:
3579:
3576:
3572:
3568:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3544:
3520:
3517:
3514:
3511:
3506:
3501:
3497:
3493:
3488:
3485:
3482:
3478:
3474:
3471:
3468:
3464:
3459:
3455:
3450:
3447:
3444:
3440:
3436:
3433:
3430:
3427:
3422:
3418:
3414:
3411:
3407:
3400:
3395:
3392:
3389:
3385:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3323:
3320:
3299:
3296:
3293:
3290:
3286:
3281:
3278:
3275:
3272:
3267:
3262:
3258:
3229:
3223:
3222:
3210:
3207:
3203:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3168:
3150:
3137:
3108:
3095:
3080:
3077:
3050:
3049:
3038:
3035:
3030:
3026:
3022:
3019:
3016:
3011:
3007:
3003:
2998:
2994:
2990:
2987:
2973:
2972:
2961:
2958:
2953:
2949:
2945:
2940:
2936:
2932:
2929:
2926:
2923:
2918:
2915:
2912:
2908:
2904:
2901:
2898:
2893:
2889:
2885:
2825:
2824:Generalization
2822:
2808:
2805:
2802:
2799:
2772:
2769:
2766:
2763:
2760:
2740:
2737:
2734:
2731:
2707:
2702:
2699:
2696:
2688:
2686:
2683:
2682:
2679:
2676:
2673:
2665:
2663:
2660:
2659:
2657:
2652:
2649:
2646:
2643:
2640:
2620:
2596:
2593:
2590:
2587:
2560:
2557:
2554:
2550:
2547:
2526:
2523:
2520:
2517:
2497:
2477:
2474:
2471:
2468:
2448:
2428:
2408:
2399:plane and the
2388:
2368:
2348:
2328:
2308:
2288:
2285:
2282:
2279:
2259:
2256:
2253:
2250:
2230:
2210:
2190:
2159:
2156:
2153:
2150:
2130:
2127:
2124:
2121:
2101:
2089:
2086:
2026:
2023:
1992:
1989:
1988:
1987:
1976:
1973:
1970:
1967:
1964:
1960:
1953:
1949:
1943:
1938:
1935:
1931:
1927:
1923:
1918:
1914:
1910:
1906:
1903:
1865:
1864:
1853:
1850:
1847:
1844:
1840:
1835:
1832:
1829:
1826:
1821:
1816:
1813:
1809:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1743:
1742:
1731:
1728:
1724:
1719:
1716:
1713:
1709:
1706:
1702:
1699:
1696:
1693:
1688:
1683:
1680:
1676:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1616:
1610:
1606:
1603:
1600:
1597:
1593:
1587:
1583:
1580:
1570:expected value
1538:
1535:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1495:
1490:
1487:
1484:
1481:
1476:
1471:
1467:
1446:
1426:
1406:
1403:
1400:
1397:
1394:
1372:
1367:
1364:
1361:
1353:
1351:
1348:
1347:
1344:
1341:
1338:
1330:
1328:
1325:
1324:
1322:
1317:
1314:
1311:
1308:
1305:
1283:
1279:
1274:
1271:
1268:
1264:
1261:
1257:
1254:
1251:
1248:
1243:
1238:
1234:
1230:
1227:
1224:
1221:
1218:
1214:
1209:
1206:
1203:
1200:
1195:
1190:
1186:
1165:
1162:
1159:
1155:
1152:
1131:
1128:
1125:
1122:
1102:
1099:
1096:
1093:
1090:
1081:is bounded on
1070:
1067:
1064:
1061:
1002:
999:
996:
993:
989:
984:
981:
978:
975:
970:
965:
961:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
887:
882:
879:
876:
873:
868:
863:
859:
840:
837:
836:
835:
823:
820:
817:
813:
809:
806:
803:
800:
797:
794:
791:
788:
785:
782:
778:
760:
718:
686:
658:
638:
618:
613:
610:
607:
603:
599:
594:
590:
586:
566:
544:
540:
528:
527:
515:
511:
506:
502:
498:
495:
492:
489:
484:
481:
478:
474:
470:
467:
463:
459:
454:
450:
446:
443:
438:
435:
432:
427:
424:
421:
417:
413:
410:
407:
404:
401:
398:
395:
392:
389:
366:
351:
350:
339:
336:
333:
330:
325:
321:
317:
314:
311:
306:
302:
298:
293:
289:
285:
282:
279:
276:
273:
251:
248:
245:
242:
239:
219:
205:
204:
193:
190:
187:
184:
181:
177:
172:
169:
166:
163:
158:
153:
150:
147:
143:
130:is denoted by
119:
99:
96:
93:
90:
87:
67:
48:
45:
31:, named after
15:
9:
6:
4:
3:
2:
6421:
6410:
6407:
6405:
6402:
6401:
6399:
6384:
6383:
6374:
6373:
6370:
6364:
6361:
6359:
6356:
6354:
6351:
6349:
6346:
6344:
6341:
6339:
6336:
6334:
6331:
6329:
6326:
6324:
6321:
6319:
6316:
6314:
6311:
6309:
6306:
6304:
6301:
6299:
6296:
6294:
6291:
6289:
6286:
6284:
6281:
6279:
6276:
6274:
6271:
6269:
6266:
6264:
6261:
6259:
6256:
6254:
6251:
6249:
6246:
6244:
6241:
6239:
6236:
6234:
6231:
6229:
6226:
6224:
6221:
6219:
6216:
6214:
6211:
6209:
6206:
6204:
6201:
6199:
6196:
6194:
6191:
6189:
6186:
6184:
6181:
6179:
6176:
6175:
6172:
6168:
6161:
6156:
6154:
6149:
6147:
6142:
6141:
6138:
6124:
6121:
6119:
6116:
6115:
6113:
6111:
6108:
6106:
6103:
6101:
6098:
6096:
6093:
6091:
6090:Basel problem
6088:
6087:
6085:
6083:Miscellaneous
6081:
6075:
6072:
6070:
6067:
6065:
6062:
6060:
6057:
6056:
6054:
6052:
6048:
6042:
6039:
6037:
6034:
6032:
6029:
6025:
6022:
6020:
6017:
6016:
6014:
6012:
6009:
6007:
6004:
6003:
6001:
5999:
5995:
5989:
5986:
5982:
5979:
5977:
5974:
5973:
5972:
5969:
5967:
5964:
5962:
5959:
5957:
5954:
5952:
5949:
5947:
5944:
5942:
5939:
5937:
5934:
5932:
5929:
5927:
5924:
5922:
5919:
5917:
5914:
5910:
5907:
5905:
5902:
5900:
5899:Trigonometric
5897:
5896:
5895:
5892:
5891:
5889:
5883:
5877:
5874:
5872:
5869:
5867:
5864:
5862:
5859:
5857:
5854:
5852:
5849:
5847:
5844:
5842:
5839:
5837:
5836:Haar integral
5834:
5832:
5829:
5827:
5824:
5822:
5819:
5817:
5814:
5812:
5809:
5807:
5804:
5802:
5799:
5798:
5796:
5790:
5786:
5779:
5774:
5772:
5767:
5765:
5760:
5759:
5756:
5747:
5742:
5738:
5734:
5730:
5725:
5721:
5719:0-8176-4073-8
5715:
5710:
5709:
5702:
5698:
5694:
5690:
5686:
5682:
5678:
5674:
5670:
5668:0-486-63519-8
5664:
5660:
5656:
5652:
5647:
5643:
5638:
5634:
5632:0-486-66289-6
5628:
5624:
5619:
5615:
5611:
5610:
5604:
5593:
5589:
5585:
5581:
5577:
5573:
5566:
5561:
5557:
5555:0-486-61226-0
5551:
5546:
5545:
5539:
5535:
5531:
5527:
5521:
5517:
5513:
5509:
5505:
5501:
5497:
5493:
5489:
5485:
5481:
5477:
5473:
5469:
5465:
5461:
5457:
5453:
5449:
5445:
5441:
5440:
5434:
5432:
5426:
5425:
5419:
5415:
5409:
5401:
5397:
5393:
5389:
5385:
5381:
5376:
5375:
5364:
5360:
5354:
5347:
5346:Graves (1946)
5342:
5335:
5331:
5326:
5319:
5314:
5307:
5301:
5295:
5290:
5283:
5278:
5271:
5266:
5259:
5254:
5247:
5241:
5234:
5229:
5222:
5217:
5210:
5205:
5201:
5193:
5176:
5170:
5147:
5141:
5118:
5112:
5092:
5071:
5068:
5046:
5043:
5019:
5013:
5007:
5004:
4997:
4991:
4985:
4982:
4975:
4972:
4967:
4960:
4954:
4951:
4944:
4938:
4930:
4924:
4916:
4910:
4906:
4898:
4897:
4896:
4879:
4876:
4873:
4867:
4861:
4855:
4852:
4846:
4840:
4820:
4797:
4791:
4788:
4782:
4776:
4750:
4744:
4741:
4738:
4712:
4706:
4683:
4680:
4677:
4651:
4645:
4636:
4620:
4617:
4613:
4609:
4606:
4586:
4577:
4557:
4549:
4546:
4543:
4540:
4534:
4528:
4522:
4513:
4504:
4487:
4475:
4469:
4461:
4455:
4447:
4441:
4437:
4433:
4427:
4421:
4409:
4403:
4398:
4393:
4389:
4381:
4380:
4379:
4377:
4375:
4370:
4351:
4348:
4345:
4336:
4330:
4324:
4310:
4296:
4293:
4287:
4281:
4267:
4265:
4261:
4258:could be the
4245:
4237:
4221:
4212:
4210:
4194:
4186:
4170:
4161:
4147:
4124:
4112:
4105:
4102:
4095:
4089:
4084:
4079:
4075:
4071:
4065:
4059:
4047:
4041:
4036:
4031:
4027:
4019:
4018:
4017:
3993:
3974:
3968:
3956:
3952:
3926:
3923:
3913:
3910:
3907:
3904:
3901:
3895:
3892:
3886:
3883:
3880:
3877:
3874:
3868:
3859:
3850:
3844:
3841:
3829:
3828:
3827:
3825:
3821:
3818:Furthermore,
3802:
3799:
3796:
3790:
3787:
3784:
3781:
3778:
3772:
3769:
3763:
3760:
3757:
3754:
3751:
3745:
3738:
3737:
3736:
3734:
3730:
3726:
3721:
3708:
3702:
3696:
3686:
3682:
3678:
3673:
3670:
3667:
3663:
3656:
3653:
3635:
3632:
3629:
3625:
3618:
3615:
3607:
3603:
3596:
3585:
3580:
3577:
3574:
3570:
3566:
3560:
3557:
3554:
3551:
3548:
3542:
3534:
3531:
3515:
3509:
3499:
3495:
3491:
3486:
3483:
3480:
3476:
3469:
3466:
3448:
3445:
3442:
3438:
3431:
3428:
3420:
3416:
3409:
3398:
3393:
3390:
3387:
3383:
3379:
3373:
3370:
3367:
3364:
3361:
3355:
3347:
3345:
3341:
3337:
3333:
3329:
3319:
3317:
3313:
3294:
3288:
3276:
3270:
3265:
3260:
3256:
3246:
3244:
3240:
3235:
3232:
3228:
3208:
3205:
3197:
3194:
3188:
3185:
3182:
3179:
3176:
3170:
3158:
3157:
3156:
3153:
3149:
3146:that refines
3145:
3140:
3136:
3132:
3128:
3124:
3120:
3116:
3111:
3107:
3103:
3098:
3094:
3090:
3086:
3076:
3074:
3070:
3066:
3061:
3059:
3055:
3036:
3033:
3028:
3024:
3020:
3017:
3014:
3009:
3005:
3001:
2996:
2992:
2988:
2985:
2978:
2977:
2976:
2956:
2951:
2938:
2934:
2927:
2924:
2916:
2913:
2910:
2906:
2899:
2891:
2887:
2876:
2875:
2874:
2872:
2868:
2863:
2859:
2854:
2850:
2846:
2841:
2839:
2835:
2831:
2803:
2797:
2788:
2784:
2770:
2764:
2758:
2735:
2729:
2720:
2700:
2697:
2694:
2684:
2677:
2674:
2671:
2661:
2655:
2650:
2644:
2638:
2618:
2591:
2585:
2576:
2572:
2555:
2548:
2545:
2521:
2515:
2495:
2472:
2466:
2446:
2426:
2406:
2386:
2366:
2346:
2326:
2306:
2283:
2277:
2254:
2248:
2228:
2208:
2188:
2175:
2171:
2154:
2148:
2125:
2119:
2099:
2085:
2083:
2082:discontinuity
2079:
2075:
2071:
2067:
2063:
2058:
2056:
2052:
2048:
2044:
2040:
2036:
2032:
2022:
2020:
2015:
2013:
2009:
2006:
2002:
1998:
1974:
1968:
1962:
1951:
1947:
1933:
1929:
1925:
1921:
1916:
1912:
1908:
1904:
1894:
1893:
1892:
1890:
1886:
1882:
1878:
1874:
1870:
1848:
1842:
1830:
1824:
1811:
1807:
1803:
1794:
1788:
1782:
1772:
1771:
1770:
1768:
1764:
1760:
1756:
1752:
1748:
1729:
1726:
1714:
1707:
1704:
1697:
1691:
1678:
1674:
1670:
1661:
1655:
1649:
1639:
1638:
1637:
1635:
1631:
1614:
1608:
1601:
1595:
1591:
1585:
1581:
1571:
1567:
1563:
1559:
1555:
1552:
1548:
1544:
1534:
1518:
1512:
1509:
1503:
1497:
1485:
1479:
1474:
1469:
1465:
1444:
1424:
1404:
1401:
1398:
1395:
1392:
1365:
1362:
1359:
1349:
1342:
1339:
1336:
1326:
1320:
1315:
1309:
1303:
1294:
1281:
1269:
1262:
1259:
1252:
1246:
1241:
1236:
1232:
1228:
1222:
1216:
1204:
1198:
1193:
1188:
1184:
1160:
1153:
1150:
1126:
1120:
1097:
1094:
1091:
1065:
1059:
1050:
1046:
1042:
1037:
1033:
1029:
1025:
1021:
1016:
1013:
997:
991:
979:
973:
968:
963:
959:
955:
949:
943:
937:
931:
928:
922:
916:
910:
904:
901:
895:
889:
877:
871:
866:
861:
857:
848:
846:
821:
818:
815:
807:
804:
798:
795:
792:
789:
786:
780:
768:
767:
766:
763:
759:
755:
751:
747:
743:
739:
735:
730:
716:
708:
704:
700:
684:
676:
672:
656:
636:
611:
608:
605:
601:
597:
592:
588:
564:
542:
538:
513:
504:
500:
493:
490:
482:
479:
476:
472:
465:
461:
452:
448:
441:
436:
433:
430:
425:
422:
419:
415:
411:
405:
402:
399:
396:
393:
387:
380:
379:
378:
364:
356:
337:
331:
328:
323:
319:
315:
312:
309:
304:
300:
296:
291:
287:
283:
280:
274:
271:
264:
263:
262:
246:
243:
240:
217:
210:
191:
185:
179:
167:
161:
156:
151:
148:
145:
141:
133:
132:
131:
117:
94:
91:
88:
65:
58:
54:
44:
42:
38:
34:
30:
26:
22:
6380:
6357:
6248:Riemann form
6059:Itô integral
5894:Substitution
5885:Integration
5870:
5736:
5732:
5707:
5688:
5684:
5650:
5641:
5622:
5613:
5607:
5595:. Retrieved
5575:
5571:
5543:
5515:
5491:
5484:Hille, Einar
5443:
5437:
5423:
5408:cite journal
5383:
5379:
5353:
5341:
5333:
5325:
5313:
5300:
5289:
5277:
5270:Rudin (1964)
5265:
5253:
5248:for details.
5240:
5233:Young (1936)
5228:
5216:
5204:
5034:
4637:
4599:, or to use
4574:
4502:
4373:
4316:
4273:
4263:
4235:
4213:
4162:
4139:
3960:
3954:
3950:
3823:
3819:
3817:
3732:
3728:
3724:
3722:
3535:
3532:
3348:
3343:
3339:
3335:
3331:
3328:Darboux sums
3325:
3322:Darboux sums
3315:
3311:
3247:
3243:directed set
3236:
3230:
3226:
3224:
3151:
3147:
3143:
3138:
3134:
3130:
3126:
3125:is a number
3122:
3118:
3114:
3109:
3105:
3104:arises from
3101:
3096:
3092:
3088:
3084:
3082:
3065:Itô integral
3062:
3051:
2974:
2870:
2866:
2861:
2857:
2853:Banach space
2848:
2844:
2842:
2827:
2721:
2610:
2180:
2091:
2077:
2073:
2069:
2065:
2061:
2059:
2054:
2050:
2046:
2042:
2034:
2030:
2028:
2016:
2007:
2005:Banach space
1994:
1888:
1880:
1876:
1872:
1868:
1866:
1765:(again, the
1761:fails to be
1758:
1754:
1750:
1746:
1744:
1636:and we have
1633:
1629:
1565:
1553:
1542:
1540:
1295:
1051:
1044:
1040:
1035:
1031:
1023:
1019:
1017:
1014:
849:
847:in the form
842:
761:
757:
753:
749:
745:
741:
737:
733:
731:
677:. Typically
674:
670:
529:
352:
206:
50:
24:
18:
6273:Riemann sum
5909:Weierstrass
4833:-axis, and
4371:, called a
2060:A function
1556:that has a
21:mathematics
6398:Categories
6024:incomplete
5887:techniques
5597:2 November
5431:HathiTrust
5372:References
5163:intersect
3735:such that
3056:, via the
3054:semigroups
2508:for which
2001:dual space
839:Properties
748:with mesh(
675:integrator
557:is in the
209:partitions
5794:integrals
5792:Types of
5785:Integrals
5691:: 1–122.
5540:(1975) .
5460:0002-9890
4968:∫
4907:∫
4877:−
4618:−
4438:∫
4390:∫
4313:Rectifier
4076:∫
4028:∫
3893:−
3857:→
3845:
3800:ε
3770:−
3671:−
3657:∈
3633:−
3616:−
3571:∑
3484:−
3470:∈
3446:−
3429:−
3384:∑
3257:∫
3209:ε
3195:−
3021:≤
3018:⋯
3015:≤
3002:≤
2960:∞
2948:‖
2925:−
2914:−
2897:‖
2888:∑
2675:≤
1942:∞
1937:∞
1934:−
1930:∫
1905:
1883:, if the
1867:holds if
1820:∞
1815:∞
1812:−
1808:∫
1783:
1687:∞
1682:∞
1679:−
1675:∫
1650:
1582:
1466:∫
1340:≤
1233:∫
1185:∫
960:∫
956:−
929:−
858:∫
819:ε
805:−
671:integrand
491:−
434:−
416:∑
313:⋯
142:∫
6382:Category
6114:Volumes
6019:complete
5916:By parts
5679:(1894).
5578:: 1–11.
5490:(1974).
5085:are the
5072:′
5047:′
4986:′
4976:′
4106:′
3961:Given a
2838:improper
2691:if
2668:if
2549:′
1708:′
1356:if
1333:if
1263:′
1154:′
1049: .
699:monotone
673:and the
53:integral
6118:Washers
5697:1344720
5655:Bibcode
5592:2307181
5504:0423094
5476:1524276
5468:2302540
5400:2322483
5223:, §3.3.
4183:is the
3241:on the
2181:If the
2003:of the
1545:is the
1457:, then
6123:Shells
5716:
5695:
5665:
5629:
5590:
5552:
5522:
5502:
5474:
5466:
5458:
5398:
5035:where
4964:
4958:
4813:, the
4376:(ReLU)
4236:almost
3089:refine
2855:. If
2141:, and
2037:is of
1885:moment
1564:, and
1385:where
1047:> 1
705:) and
530:where
23:, the
5904:Euler
5588:JSTOR
5568:(PDF)
5464:JSTOR
5396:JSTOR
5361:from
5196:Notes
5134:and
3994:over
3234:in .
3087:that
2611:When
1549:of a
1417:, if
765:in ,
55:of a
5714:ISBN
5689:VIII
5663:ISBN
5627:ISBN
5599:2010
5550:ISBN
5520:ISBN
5456:ISSN
5429:via
5414:link
5244:See
5060:and
3842:mesh
3797:<
3314:and
3206:<
3063:The
2957:<
2698:>
2076:and
2045:and
1402:<
1396:<
1363:>
1030:and
816:<
649:and
355:mesh
316:<
310:<
297:<
35:and
5741:doi
5580:doi
5448:doi
5388:doi
4340:max
3838:lim
3650:inf
3463:sup
3346:by
3117:of
2884:sup
1873:any
1871:is
1541:If
1052:If
1034:is
1022:is
19:In
6400::
5737:67
5735:.
5731:.
5693:MR
5687:.
5683:.
5661:.
5614:49
5612:.
5586:.
5576:59
5574:.
5570:.
5536:;
5500:MR
5498:.
5486:;
5472:MR
5470:.
5462:.
5454:.
5444:45
5442:.
5410:}}
5406:{{
5394:.
5384:95
5382:.
5192:.
4309:.
4266:.
4211:.
3927:0.
3155:,
3075:.
3060:.
2112:,
1887:E(
1113:,
1043:+
6159:e
6152:t
6145:v
5777:e
5770:t
5763:v
5749:.
5743::
5722:.
5699:.
5671:.
5657::
5635:.
5616:.
5601:.
5582::
5558:.
5528:.
5506:.
5478:.
5450::
5416:)
5402:.
5390::
5336:.
5320:.
5235:.
5180:)
5177:x
5174:(
5171:f
5151:)
5148:y
5145:(
5142:b
5122:)
5119:y
5116:(
5113:a
5093:x
5069:b
5044:a
5020:,
5017:)
5014:x
5011:(
5008:g
5005:d
5001:)
4998:x
4995:(
4992:f
4983:b
4973:a
4961:=
4955:x
4952:d
4948:)
4945:x
4942:(
4939:f
4934:)
4931:y
4928:(
4925:b
4920:)
4917:y
4914:(
4911:a
4883:)
4880:a
4874:b
4871:(
4868:+
4865:)
4862:y
4859:(
4856:a
4853:=
4850:)
4847:y
4844:(
4841:b
4821:x
4801:)
4798:y
4795:(
4792:a
4789:,
4786:)
4783:x
4780:(
4777:f
4757:)
4754:)
4751:x
4748:(
4745:f
4742:,
4739:x
4736:(
4716:)
4713:y
4710:(
4707:a
4687:]
4684:b
4681:,
4678:a
4675:[
4655:)
4652:x
4649:(
4646:f
4621:1
4614:h
4610:=
4607:g
4587:h
4558:3
4554:)
4550:8
4547:+
4544:x
4541:2
4538:(
4535:=
4532:)
4529:x
4526:(
4523:f
4488:x
4484:d
4479:)
4476:x
4473:(
4470:f
4465:)
4462:b
4459:(
4456:g
4451:)
4448:a
4445:(
4442:g
4434:=
4431:)
4428:x
4425:(
4422:g
4418:d
4413:)
4410:x
4407:(
4404:f
4399:b
4394:a
4355:}
4352:x
4349:,
4346:0
4343:{
4337:=
4334:)
4331:x
4328:(
4325:g
4297:x
4294:=
4291:)
4288:x
4285:(
4282:g
4264:g
4246:g
4222:g
4195:g
4171:g
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