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The concept of a convex set (i.e., a set containing the segment connecting any two of its points) had repeatedly been placed at the center of economic theory before 1964. It appeared in a new light with the introduction of integration theory in the study of
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542:
2203:. Mathematics and its Applications (East European Series). Vol. 29 (Translated from the Polish by Ewa Bednarczuk ed.). Dordrecht; Warsaw: D. Reidel Publishing Co.; PWNâPolish Scientific Publishers. pp. xvi+524.
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2465:. Vol. 81, number 1, no. Presidential address delivered at the 103rd meeting of the American Economic Association, 29 December 1990, Washington, DC. pp. 1â7.
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2267:. Research paper. Vol. 892. Palo Alto, CA: Graduate School of Business, Stanford University. pp. 30â35. (Draft of articles for the first edition of
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economic competition: If one associates with every agent of an economy an arbitrary set in the commodity space and
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over a collection of insignificant agents is an insight that economic theory owes ... to integration theory.
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2513:
Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points".
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2489:. Mathematics in Science and Engineering. Vol. 56. New YorkâLondon: Academic Press. pp. viii+136.
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2709:. Mathematical Surveys. Vol. 15. Providence, R.I: American Mathematical Society. pp. xiii+322.
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744:{\displaystyle \lim _{n\to \infty }\left\|\mu {\left(\bigcup _{i=n}^{\infty }A_{i}\right)}\right\|=0,}
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537:{\displaystyle \mu {\left(\bigcup _{i=1}^{\infty }A_{i}\right)}=\sum _{i=1}^{\infty }\mu (A_{i})}
2633:(1978). "A note on the core equivalence theorem: How many blocking coalitions are there?".
2256:
1670:
770:
2594:(2008). "ShapleyâFolkman theorem". In Durlauf, Steven N.; Blume, Lawrence E. (eds.).
2293:(January 1966). "Existence of competitive equilibrium in markets with a continuum of traders".
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2441:. But explanations of the ... functions of prices ... can be made to rest on the
2411:
Vind, Karl (May 1964). "Edgeworth-allocations in an exchange economy with many traders".
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values satisfying certain properties. It is a generalization of the concept of finite
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2092: â generalization of the Lebesgue integral to Banach-space valued functions
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Aumann, Robert J. (JanuaryâApril 1964). "Markets with a continuum of traders".
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Tardella, Fabio (1990). "A new proof of the
Lyapunov convexity theorem".
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2119: â Function valued in a vector space; typically a real or complex one
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Aumann, Robert J. (August 1965). "Integrals of set-valued functions".
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Both of these statements follow quite easily from the criterion (
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is declared to take values, two different outcomes are observed.
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1532:{\displaystyle |\mu |(A)=\sup \sum _{i=1}^{n}\|\mu (A_{i})\|}
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Differentiable vectorâvalued functions from
Euclidean space
2040:. In fact, the range of a non-atomic vector measure is a
2600:(Second ed.). Palgrave Macmillan. pp. 317â318.
2678:(reprint ed.). Boston–Basel–Stuttgart:
2461:(March 1991). "The Mathematization of economic theory".
2443:
convexity of sets derived by that averaging process
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Pages displaying short descriptions of redirect targets
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Functional analysis and control theory: Linear systems
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889:Consider the field of sets made up of the interval
3572:Spectral theory of ordinary differential equations
2175:. Providence, R.I: American Mathematical Society.
2064:. Lyapunov's theorem has been proved by using the
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577:It can be proved that an additive vector measure
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838:taking values respectively on the real interval
661:
2485:Hermes, Henry; LaSalle, Joseph P. (1969).
1963:is a vector measure of bounded variation, then
1620:into a finite number of disjoint sets, for all
818:Countably additive vector measures defined on
303:{\displaystyle \mu (A\cup B)=\mu (A)+\mu (B).}
3712:
3228:
2798:
2597:The New Palgrave Dictionary of Economics
949:contained in this interval. For any such set
8:
4457:RieszâMarkovâKakutani representation theorem
2487:Functional analysis and time optimal control
2439:then the resulting set is necessarily convex
2415:. Vol. 5, no. 2. pp. 165â77.
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2032:) finite-dimensional vector measure is
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2437:over a collection of insignificant agents,
2335:This paper builds on two papers by Aumann:
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2435:if one averages those individual sets
2730:, North-Holland Mathematics Studies
2269:New Palgrave Dictionary of Economics
2154:, North-Holland Mathematics Studies
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3525:Group algebra of a locally compact group
3102:No infinite-dimensional Lebesgue measure
2705:Diestel, Joe; Uhl, Jerry J. Jr. (1977).
2558:SIAM Journal on Control and Optimization
2171:Diestel, Joe; Uhl, Jerry J. Jr. (1977).
168:{\displaystyle \mu :{\mathcal {F}}\to X}
98:{\displaystyle (\Omega ,{\mathcal {F}})}
3112:Structure theorem for Gaussian measures
2137:
1338:is a countably-additive vector measure.
641:{\displaystyle (A_{i})_{i=1}^{\infty }}
383:{\displaystyle (A_{i})_{i=1}^{\infty }}
2426:
2988:infinite-dimensional Gaussian measure
1983:is countably additive if and only if
7:
4565:Applications & related
2859:Infinite-dimensional vector function
650:
2728:Vector Measures and Control Systems
2243:(July 1986). "Large economies". In
2152:Vector Measures and Control Systems
2107: â Measure with complex values
2021:In the theory of vector measures,
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62:Definitions and first consequences
25:
2926:Generalizations of the derivative
2892:Differentiation in Fréchet spaces
2764:. New York: McGraw-Hill. p.
2635:Journal of Mathematical Economics
2101: â Type of topological space
1352:The variation of a vector measure
1007:{\displaystyle \mu (A)=\chi _{A}}
4494:Lebesgue differentiation theorem
4375:Carathéodory's extension theorem
3681:
3680:
3607:Topological quantum field theory
2429:, p. 4) with this comment:
133:finitely additive vector measure
3161:Holomorphic functional calculus
1912:{\displaystyle |\mu |(\Omega )}
1873:{\displaystyle {\mathcal {F}}.}
1660:{\displaystyle {\mathcal {F}}.}
3156:Continuous functional calculus
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3403:Uniform boundedness principle
2413:International Economic Review
2068:, which has been viewed as a
822:are more general than finite
2647:10.1016/0304-4068(78)90010-1
2463:The American Economic Review
2449:in the commodity space
2425:Vind's article was noted by
2389:10.1016/0022-247X(65)90049-1
2028:states that the range of a (
866:{\displaystyle [0,\infty ),}
836:countably additive functions
414:such that their union is in
4547:PrĂ©kopaâLeindler inequality
2745:Encyclopedia of Mathematics
2085:Bochner measurable function
1344:
139:, for short) is a function
4639:
4489:Lebesgue's density theorem
3546:Invariant subspace problem
2606:10.1057/9780230226203.1518
2126:Weakly measurable function
1686:{\displaystyle \|\cdot \|}
1236:viewed as a function from
1107:viewed as a function from
786:{\displaystyle \|\cdot \|}
4623:Measures (measure theory)
4542:MinkowskiâSteiner formula
4472:
4357:Projection-valued measure
3676:
3266:
3151:Borel functional calculus
2818:topological vector spaces
2672:Cohn, Donald L. (1997) .
2199:Rolewicz, Stefan (1987).
921:together with the family
4525:Isoperimetric inequality
4504:VitaliâHahnâSaks theorem
3833:Carathéodory's criterion
3515:Spectrum of a C*-algebra
3085:Inverse function theorem
2972:Classical Wiener measure
1943:. One can prove that if
1331:{\displaystyle L^{1}(),}
947:Lebesgue measurable sets
4530:BrunnâMinkowski theorem
4399:Decomposition theorems
3612:Noncommutative geometry
3187:Convenient vector space
2738:van Dulst, D. (2001) ,
2451:obtained by aggregation
2117:Vector-valued functions
2075:of Lyapunov's theorem.
2016:
2013:is countably additive.
1919:is finite, the measure
1356:Given a vector measure
18:Lyapunov's theorem
4577:Descriptive set theory
4477:Disintegration theorem
3912:Universally measurable
3668:TomitaâTakesaki theory
3643:Approximation property
3587:Calculus of variations
3080:CameronâMartin theorem
2837:Classical Wiener space
2455:
2291:Aumann, Robert J.
2007:
2006:{\displaystyle |\mu |}
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1421:{\displaystyle |\mu |}
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175:such that for any two
169:
125:
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4379:Convergence theorems
3838:Cylindrical Ï-algebra
3663:BanachâMazur distance
3626:Generalized functions
3097:FeldmanâHĂĄjek theorem
2909:Functional derivative
2832:Abstract Wiener space
2430:
2258:Contributions to the
2066:ShapleyâFolkman lemma
2008:
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1280:{\displaystyle L^{p}}
1255:
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1229:{\displaystyle \mu ,}
1206:
1154:
1152:{\displaystyle L^{p}}
1126:
1102:
1100:{\displaystyle \mu ,}
1076:
1056:
1029:
1027:{\displaystyle \chi }
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4447:Minkowski inequality
4321:Cylinder set measure
4206:Infinite-dimensional
3821:equivalence relation
3751:Lebesgue integration
3408:Kakutani fixed-point
3393:Riesz representation
3021:Radonifying function
2962:Cylinder set measure
2854:Cylinder set measure
1987:
1976:{\displaystyle \mu }
1967:
1956:{\displaystyle \mu }
1947:
1932:{\displaystyle \mu }
1923:
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1731:{\displaystyle \mu }
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1441:{\displaystyle \mu }
1432:
1402:
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1111:
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1074:{\displaystyle \mu }
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601:
590:{\displaystyle \mu }
581:
556:
552:of the Banach space
442:
418:
394:
390:of disjoint sets in
343:
325:{\displaystyle \mu }
316:
246:
222:
202:
182:
143:
112:
73:
4618:Functional analysis
4442:Hölder's inequality
4304:of random variables
4266:Measurable function
4153:Particular measures
3742:Absolute continuity
3592:Functional calculus
3551:Mahler's conjecture
3530:Von Neumann algebra
3244:Functional analysis
3143:Functional calculus
3133:Covariance operator
3054:GelfandâPettis/Weak
3016:measurable function
2931:Hadamard derivative
2760:Functional analysis
2682:. pp. IX+373.
2592:Starr, Ross M.
1061:Depending on where
637:
379:
4582:Probability theory
3907:Transverse measure
3885:Non-measurable set
3867:Locally measurable
3617:Riemann hypothesis
3316:Topological vector
3090:NashâMoser theorem
2967:Canonical Gaussian
2914:Gateaux derivative
2897:Fréchet derivative
2734:, Amsterdam, 1976.
2631:Mas-Colell, Andreu
2158:, Amsterdam, 1976.
2062:statistical theory
2017:Lyapunov's theorem
2003:
1973:
1953:
1929:
1909:
1870:
1843:
1823:
1763:
1728:
1709:{\displaystyle X.}
1706:
1683:
1657:
1630:
1610:
1590:
1529:
1438:
1418:
1385:
1328:
1277:
1250:
1226:
1201:
1149:
1121:
1097:
1071:
1054:{\displaystyle A.}
1051:
1036:indicator function
1024:
1004:
965:{\displaystyle A,}
962:
935:
911:
863:
809:{\displaystyle X.}
806:
783:
741:
675:
638:
617:
587:
568:{\displaystyle X.}
565:
534:
428:
404:
380:
359:
334:countably additive
322:
300:
232:
208:
188:
165:
124:{\displaystyle X,}
121:
95:
4600:
4599:
4560:
4559:
4289:almost everywhere
4235:Spherical measure
4133:Strictly positive
4061:Projection-valued
3801:Almost everywhere
3774:Probability space
3694:
3693:
3597:Integral operator
3374:
3373:
3210:
3209:
3107:Sazonov's theorem
2993:Projection-valued
2756:Rudin, W (1973).
2740:"Vector measures"
2680:BirkhÀuser Verlag
2615:978-0-333-78676-5
2054:"bang–bang"
2048:). It is used in
1941:bounded variation
1939:is said to be of
1846:{\displaystyle A}
1766:{\displaystyle .}
1718:The variation of
1633:{\displaystyle A}
1613:{\displaystyle A}
877:, and the set of
765:
764:
660:
648:as above one has
312:A vector measure
211:{\displaystyle B}
191:{\displaystyle A}
16:(Redirected from
4630:
4535:Milman's reverse
4518:
4516:Lebesgue measure
4470:
3874:
3860:infimum/supremum
3781:Measurable space
3721:
3714:
3707:
3698:
3684:
3683:
3602:Jones polynomial
3520:Operator algebra
3264:
3237:
3230:
3223:
3214:
3202:Hilbert manifold
3197:Fréchet manifold
2981: like
2941:Quasi-derivative
2807:
2800:
2793:
2784:
2779:
2763:
2752:
2720:
2701:
2659:
2658:
2626:
2620:
2619:
2588:
2582:
2581:
2553:
2547:
2546:
2510:
2499:
2498:
2482:
2476:
2474:
2424:
2408:
2402:
2400:
2371:
2334:
2287:
2281:
2280:
2278:
2276:
2266:
2253:Robert B. Wilson
2237:
2231:
2230:
2196:
2187:
2186:
2168:
2159:
2145:
2122:
2095:
2090:Bochner integral
2012:
2010:
2009:
2004:
2002:
1994:
1982:
1980:
1979:
1974:
1962:
1960:
1959:
1954:
1938:
1936:
1935:
1930:
1918:
1916:
1915:
1910:
1899:
1891:
1879:
1877:
1876:
1871:
1866:
1865:
1852:
1850:
1849:
1844:
1832:
1830:
1829:
1824:
1813:
1805:
1772:
1770:
1769:
1764:
1737:
1735:
1734:
1729:
1715:
1713:
1712:
1707:
1692:
1690:
1689:
1684:
1666:
1664:
1663:
1658:
1653:
1652:
1639:
1637:
1636:
1631:
1619:
1617:
1616:
1611:
1599:
1597:
1596:
1591:
1589:
1588:
1578:
1573:
1538:
1536:
1535:
1530:
1522:
1521:
1502:
1497:
1467:
1459:
1447:
1445:
1444:
1439:
1427:
1425:
1424:
1419:
1417:
1409:
1394:
1392:
1391:
1386:
1375:
1374:
1348:) stated above.
1337:
1335:
1334:
1329:
1303:
1302:
1286:
1284:
1283:
1278:
1276:
1275:
1259:
1257:
1256:
1251:
1249:
1248:
1235:
1233:
1232:
1227:
1210:
1208:
1207:
1202:
1176:
1175:
1158:
1156:
1155:
1150:
1148:
1147:
1130:
1128:
1127:
1122:
1120:
1119:
1106:
1104:
1103:
1098:
1080:
1078:
1077:
1072:
1060:
1058:
1057:
1052:
1033:
1031:
1030:
1025:
1013:
1011:
1010:
1005:
1003:
1002:
971:
969:
968:
963:
944:
942:
941:
936:
934:
933:
920:
918:
917:
914:{\displaystyle }
912:
872:
870:
869:
864:
832:complex measures
815:
813:
812:
807:
792:
790:
789:
784:
759:
750:
748:
747:
742:
731:
727:
726:
725:
721:
720:
719:
709:
704:
674:
651:
647:
645:
644:
639:
636:
631:
616:
615:
596:
594:
593:
588:
574:
572:
571:
566:
543:
541:
540:
535:
530:
529:
513:
508:
490:
489:
485:
484:
483:
473:
468:
437:
435:
434:
429:
427:
426:
413:
411:
410:
405:
403:
402:
389:
387:
386:
381:
378:
373:
358:
357:
331:
329:
328:
323:
309:
307:
306:
301:
241:
239:
238:
233:
231:
230:
217:
215:
214:
209:
197:
195:
194:
189:
174:
172:
171:
166:
158:
157:
130:
128:
127:
122:
104:
102:
101:
96:
91:
90:
21:
4638:
4637:
4633:
4632:
4631:
4629:
4628:
4627:
4603:
4602:
4601:
4596:
4592:Spectral theory
4572:Convex analysis
4556:
4513:
4508:
4461:
4361:
4309:in distribution
4254:
4147:
3977:Logarithmically
3916:
3872:
3855:Essential range
3789:
3730:
3725:
3695:
3690:
3672:
3636:Advanced topics
3631:
3555:
3534:
3493:
3459:HilbertâSchmidt
3432:
3423:GelfandâNaimark
3370:
3320:
3255:
3241:
3211:
3206:
3177:Banach manifold
3165:
3137:
3116:
3068:
3044:Direct integral
3025:
2945:
2873:
2869:Vector calculus
2864:Matrix calculus
2820:
2811:
2776:
2755:
2737:
2717:
2707:Vector measures
2704:
2690:
2671:
2668:
2663:
2662:
2629:
2627:
2623:
2616:
2590:
2589:
2585:
2570:10.1137/0328026
2555:
2554:
2550:
2527:10.1137/1022026
2512:
2511:
2502:
2484:
2483:
2479:
2457:
2410:
2409:
2405:
2374:
2352:10.2307/1913732
2337:
2307:10.2307/1909854
2289:
2288:
2284:
2274:
2272:
2264:
2239:
2238:
2234:
2211:
2198:
2197:
2190:
2183:
2173:Vector measures
2170:
2169:
2162:
2150:, Knowles, G.,
2146:
2139:
2134:
2120:
2105:Complex measure
2093:
2081:
2019:
1985:
1984:
1965:
1964:
1945:
1944:
1921:
1920:
1882:
1881:
1855:
1854:
1835:
1834:
1775:
1774:
1740:
1739:
1720:
1719:
1695:
1694:
1693:is the norm on
1669:
1668:
1642:
1641:
1622:
1621:
1602:
1601:
1580:
1548:
1547:
1513:
1450:
1449:
1430:
1429:
1400:
1399:
1358:
1357:
1354:
1294:
1289:
1288:
1267:
1262:
1261:
1238:
1237:
1215:
1214:
1167:
1162:
1161:
1139:
1134:
1133:
1109:
1108:
1086:
1085:
1063:
1062:
1040:
1039:
1016:
1015:
994:
974:
973:
951:
950:
923:
922:
891:
890:
887:
879:complex numbers
840:
839:
828:signed measures
795:
794:
793:is the norm on
769:
768:
757:
711:
689:
685:
680:
676:
655:
654:
607:
599:
598:
579:
578:
554:
553:
521:
475:
453:
449:
440:
439:
416:
415:
392:
391:
349:
341:
340:
314:
313:
244:
243:
220:
219:
200:
199:
180:
179:
141:
140:
110:
109:
71:
70:
64:
23:
22:
15:
12:
11:
5:
4636:
4634:
4626:
4625:
4620:
4615:
4613:Control theory
4605:
4604:
4598:
4597:
4595:
4594:
4589:
4584:
4579:
4574:
4568:
4566:
4562:
4561:
4558:
4557:
4555:
4554:
4549:
4544:
4539:
4538:
4537:
4527:
4521:
4519:
4510:
4509:
4507:
4506:
4501:
4499:Sard's theorem
4496:
4491:
4486:
4485:
4484:
4482:Lifting theory
4473:
4467:
4463:
4462:
4460:
4459:
4454:
4449:
4444:
4439:
4438:
4437:
4435:FubiniâTonelli
4427:
4422:
4417:
4416:
4415:
4410:
4405:
4397:
4396:
4395:
4390:
4385:
4377:
4371:
4369:
4363:
4362:
4360:
4359:
4354:
4349:
4344:
4339:
4334:
4329:
4323:
4318:
4317:
4316:
4314:in probability
4311:
4301:
4296:
4291:
4285:
4284:
4283:
4278:
4273:
4262:
4260:
4256:
4255:
4253:
4252:
4247:
4242:
4237:
4232:
4227:
4226:
4225:
4215:
4210:
4209:
4208:
4198:
4193:
4188:
4183:
4178:
4173:
4168:
4163:
4157:
4155:
4149:
4148:
4146:
4145:
4140:
4135:
4130:
4125:
4120:
4115:
4110:
4105:
4100:
4095:
4094:
4093:
4088:
4083:
4073:
4068:
4063:
4058:
4048:
4043:
4038:
4033:
4028:
4023:
4021:Locally finite
4018:
4008:
4003:
3998:
3993:
3988:
3983:
3973:
3968:
3963:
3958:
3953:
3948:
3943:
3938:
3933:
3927:
3925:
3918:
3917:
3915:
3914:
3909:
3904:
3899:
3894:
3893:
3892:
3882:
3877:
3869:
3864:
3863:
3862:
3852:
3847:
3846:
3845:
3835:
3830:
3825:
3824:
3823:
3813:
3808:
3803:
3797:
3795:
3791:
3790:
3788:
3787:
3778:
3777:
3776:
3766:
3761:
3753:
3748:
3738:
3736:
3735:Basic concepts
3732:
3731:
3728:Measure theory
3726:
3724:
3723:
3716:
3709:
3701:
3692:
3691:
3689:
3688:
3677:
3674:
3673:
3671:
3670:
3665:
3660:
3655:
3653:Choquet theory
3650:
3645:
3639:
3637:
3633:
3632:
3630:
3629:
3619:
3614:
3609:
3604:
3599:
3594:
3589:
3584:
3579:
3574:
3569:
3563:
3561:
3557:
3556:
3554:
3553:
3548:
3542:
3540:
3536:
3535:
3533:
3532:
3527:
3522:
3517:
3512:
3507:
3505:Banach algebra
3501:
3499:
3495:
3494:
3492:
3491:
3486:
3481:
3476:
3471:
3466:
3461:
3456:
3451:
3446:
3440:
3438:
3434:
3433:
3431:
3430:
3428:BanachâAlaoglu
3425:
3420:
3415:
3410:
3405:
3400:
3395:
3390:
3384:
3382:
3376:
3375:
3372:
3371:
3369:
3368:
3363:
3358:
3356:Locally convex
3353:
3339:
3334:
3328:
3326:
3322:
3321:
3319:
3318:
3313:
3308:
3303:
3298:
3293:
3288:
3283:
3278:
3273:
3267:
3261:
3257:
3256:
3242:
3240:
3239:
3232:
3225:
3217:
3208:
3207:
3205:
3204:
3199:
3194:
3192:Choquet theory
3189:
3184:
3173:
3171:
3167:
3166:
3164:
3163:
3158:
3153:
3147:
3145:
3139:
3138:
3136:
3135:
3130:
3124:
3122:
3118:
3117:
3115:
3114:
3109:
3104:
3099:
3094:
3093:
3092:
3082:
3076:
3074:
3070:
3069:
3067:
3066:
3061:
3056:
3051:
3046:
3041:
3035:
3033:
3027:
3026:
3024:
3023:
3018:
3002:
3001:
3000:
2995:
2990:
2976:
2975:
2974:
2969:
2959:
2953:
2951:
2947:
2946:
2944:
2943:
2938:
2933:
2928:
2923:
2922:
2921:
2911:
2906:
2905:
2904:
2894:
2889:
2883:
2881:
2875:
2874:
2872:
2871:
2866:
2861:
2856:
2851:
2846:
2841:
2840:
2839:
2828:
2826:
2825:Basic concepts
2822:
2821:
2812:
2810:
2809:
2802:
2795:
2787:
2781:
2780:
2774:
2753:
2735:
2726:, Knowles, G,
2721:
2715:
2702:
2688:
2675:Measure theory
2667:
2664:
2661:
2660:
2641:(3): 207â215.
2621:
2614:
2583:
2564:(2): 478â481.
2548:
2521:(2): 172â185.
2500:
2477:
2459:Debreu, GĂ©rard
2403:
2346:(1â2): 39â50.
2282:
2245:David M. Kreps
2232:
2209:
2188:
2181:
2160:
2136:
2135:
2133:
2130:
2129:
2128:
2123:
2114:
2111:Signed measure
2108:
2102:
2096:
2087:
2080:
2077:
2058:control theory
2018:
2015:
2001:
1997:
1993:
1972:
1952:
1928:
1908:
1905:
1902:
1898:
1894:
1890:
1869:
1864:
1842:
1822:
1819:
1816:
1812:
1808:
1804:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1773:It holds that
1762:
1759:
1756:
1753:
1750:
1747:
1727:
1705:
1702:
1682:
1679:
1676:
1656:
1651:
1629:
1609:
1587:
1583:
1577:
1572:
1569:
1566:
1562:
1558:
1555:
1528:
1525:
1520:
1516:
1512:
1509:
1506:
1501:
1496:
1493:
1490:
1486:
1482:
1479:
1476:
1473:
1470:
1466:
1462:
1458:
1448:is defined as
1437:
1416:
1412:
1408:
1384:
1381:
1378:
1373:
1368:
1365:
1353:
1350:
1340:
1339:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1301:
1297:
1274:
1270:
1247:
1225:
1222:
1212:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1174:
1170:
1146:
1142:
1118:
1096:
1093:
1070:
1050:
1047:
1023:
1001:
997:
993:
990:
987:
984:
981:
961:
958:
932:
910:
907:
904:
901:
898:
886:
883:
862:
859:
856:
853:
850:
847:
820:sigma-algebras
805:
802:
782:
779:
776:
763:
762:
753:
751:
740:
737:
734:
730:
724:
718:
714:
708:
703:
700:
697:
693:
688:
683:
679:
673:
670:
667:
663:
635:
630:
627:
624:
620:
614:
610:
606:
586:
564:
561:
533:
528:
524:
520:
517:
512:
507:
504:
501:
497:
493:
488:
482:
478:
472:
467:
464:
461:
457:
452:
447:
438:it holds that
425:
401:
377:
372:
369:
366:
362:
356:
352:
348:
321:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
229:
207:
187:
164:
161:
156:
151:
148:
120:
117:
94:
89:
84:
81:
78:
63:
60:
51:, which takes
41:family of sets
33:vector measure
24:
14:
13:
10:
9:
6:
4:
3:
2:
4635:
4624:
4621:
4619:
4616:
4614:
4611:
4610:
4608:
4593:
4590:
4588:
4587:Real analysis
4585:
4583:
4580:
4578:
4575:
4573:
4570:
4569:
4567:
4563:
4553:
4550:
4548:
4545:
4543:
4540:
4536:
4533:
4532:
4531:
4528:
4526:
4523:
4522:
4520:
4517:
4511:
4505:
4502:
4500:
4497:
4495:
4492:
4490:
4487:
4483:
4480:
4479:
4478:
4475:
4474:
4471:
4468:
4466:Other results
4464:
4458:
4455:
4453:
4452:RadonâNikodym
4450:
4448:
4445:
4443:
4440:
4436:
4433:
4432:
4431:
4428:
4426:
4425:Fatou's lemma
4423:
4421:
4418:
4414:
4411:
4409:
4406:
4404:
4401:
4400:
4398:
4394:
4391:
4389:
4386:
4384:
4381:
4380:
4378:
4376:
4373:
4372:
4370:
4368:
4364:
4358:
4355:
4353:
4350:
4348:
4345:
4343:
4340:
4338:
4335:
4333:
4330:
4328:
4324:
4322:
4319:
4315:
4312:
4310:
4307:
4306:
4305:
4302:
4300:
4297:
4295:
4292:
4290:
4287:Convergence:
4286:
4282:
4279:
4277:
4274:
4272:
4269:
4268:
4267:
4264:
4263:
4261:
4257:
4251:
4248:
4246:
4243:
4241:
4238:
4236:
4233:
4231:
4228:
4224:
4221:
4220:
4219:
4216:
4214:
4211:
4207:
4204:
4203:
4202:
4199:
4197:
4194:
4192:
4189:
4187:
4184:
4182:
4179:
4177:
4174:
4172:
4169:
4167:
4164:
4162:
4159:
4158:
4156:
4154:
4150:
4144:
4141:
4139:
4136:
4134:
4131:
4129:
4126:
4124:
4121:
4119:
4116:
4114:
4111:
4109:
4106:
4104:
4101:
4099:
4096:
4092:
4091:Outer regular
4089:
4087:
4086:Inner regular
4084:
4082:
4081:Borel regular
4079:
4078:
4077:
4074:
4072:
4069:
4067:
4064:
4062:
4059:
4057:
4053:
4049:
4047:
4044:
4042:
4039:
4037:
4034:
4032:
4029:
4027:
4024:
4022:
4019:
4017:
4013:
4009:
4007:
4004:
4002:
3999:
3997:
3994:
3992:
3989:
3987:
3984:
3982:
3978:
3974:
3972:
3969:
3967:
3964:
3962:
3959:
3957:
3954:
3952:
3949:
3947:
3944:
3942:
3939:
3937:
3934:
3932:
3929:
3928:
3926:
3924:
3919:
3913:
3910:
3908:
3905:
3903:
3900:
3898:
3895:
3891:
3888:
3887:
3886:
3883:
3881:
3878:
3876:
3870:
3868:
3865:
3861:
3858:
3857:
3856:
3853:
3851:
3848:
3844:
3841:
3840:
3839:
3836:
3834:
3831:
3829:
3826:
3822:
3819:
3818:
3817:
3814:
3812:
3809:
3807:
3804:
3802:
3799:
3798:
3796:
3792:
3786:
3782:
3779:
3775:
3772:
3771:
3770:
3769:Measure space
3767:
3765:
3762:
3760:
3758:
3754:
3752:
3749:
3747:
3743:
3740:
3739:
3737:
3733:
3729:
3722:
3717:
3715:
3710:
3708:
3703:
3702:
3699:
3687:
3679:
3678:
3675:
3669:
3666:
3664:
3661:
3659:
3658:Weak topology
3656:
3654:
3651:
3649:
3646:
3644:
3641:
3640:
3638:
3634:
3627:
3623:
3620:
3618:
3615:
3613:
3610:
3608:
3605:
3603:
3600:
3598:
3595:
3593:
3590:
3588:
3585:
3583:
3582:Index theorem
3580:
3578:
3575:
3573:
3570:
3568:
3565:
3564:
3562:
3558:
3552:
3549:
3547:
3544:
3543:
3541:
3539:Open problems
3537:
3531:
3528:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3508:
3506:
3503:
3502:
3500:
3496:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3460:
3457:
3455:
3452:
3450:
3447:
3445:
3442:
3441:
3439:
3435:
3429:
3426:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3385:
3383:
3381:
3377:
3367:
3364:
3362:
3359:
3357:
3354:
3351:
3347:
3343:
3340:
3338:
3335:
3333:
3330:
3329:
3327:
3323:
3317:
3314:
3312:
3309:
3307:
3304:
3302:
3299:
3297:
3294:
3292:
3289:
3287:
3284:
3282:
3279:
3277:
3274:
3272:
3269:
3268:
3265:
3262:
3258:
3253:
3249:
3245:
3238:
3233:
3231:
3226:
3224:
3219:
3218:
3215:
3203:
3200:
3198:
3195:
3193:
3190:
3188:
3185:
3182:
3178:
3175:
3174:
3172:
3168:
3162:
3159:
3157:
3154:
3152:
3149:
3148:
3146:
3144:
3140:
3134:
3131:
3129:
3126:
3125:
3123:
3119:
3113:
3110:
3108:
3105:
3103:
3100:
3098:
3095:
3091:
3088:
3087:
3086:
3083:
3081:
3078:
3077:
3075:
3071:
3065:
3062:
3060:
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3037:
3036:
3034:
3032:
3028:
3022:
3019:
3017:
3014:
3010:
3006:
3003:
2999:
2996:
2994:
2991:
2989:
2986:
2985:
2984:
2983:set functions
2980:
2977:
2973:
2970:
2968:
2965:
2964:
2963:
2960:
2958:
2957:Besov measure
2955:
2954:
2952:
2950:Measurability
2948:
2942:
2939:
2937:
2934:
2932:
2929:
2927:
2924:
2920:
2917:
2916:
2915:
2912:
2910:
2907:
2903:
2900:
2899:
2898:
2895:
2893:
2890:
2888:
2885:
2884:
2882:
2880:
2876:
2870:
2867:
2865:
2862:
2860:
2857:
2855:
2852:
2850:
2849:Convex series
2847:
2845:
2844:Bochner space
2842:
2838:
2835:
2834:
2833:
2830:
2829:
2827:
2823:
2819:
2815:
2808:
2803:
2801:
2796:
2794:
2789:
2788:
2785:
2777:
2775:9780070542259
2771:
2767:
2762:
2761:
2754:
2751:
2747:
2746:
2741:
2736:
2733:
2729:
2725:
2722:
2718:
2716:0-8218-1515-6
2712:
2708:
2703:
2699:
2695:
2691:
2689:3-7643-3003-1
2685:
2681:
2677:
2676:
2670:
2669:
2665:
2656:
2652:
2648:
2644:
2640:
2636:
2632:
2625:
2622:
2617:
2611:
2607:
2603:
2599:
2598:
2593:
2587:
2584:
2579:
2575:
2571:
2567:
2563:
2559:
2552:
2549:
2544:
2540:
2536:
2532:
2528:
2524:
2520:
2516:
2509:
2507:
2505:
2501:
2496:
2492:
2488:
2481:
2478:
2475:
2472:
2468:
2464:
2460:
2454:
2452:
2448:
2444:
2440:
2436:
2428:
2422:
2418:
2414:
2407:
2404:
2401:
2398:
2394:
2390:
2386:
2382:
2378:
2372:
2369:
2365:
2361:
2357:
2353:
2349:
2345:
2341:
2332:
2328:
2324:
2320:
2316:
2312:
2308:
2304:
2300:
2296:
2292:
2286:
2283:
2270:
2263:
2262:
2259:
2254:
2250:
2246:
2242:
2241:Roberts, John
2236:
2233:
2228:
2224:
2220:
2216:
2212:
2210:90-277-2186-6
2206:
2202:
2195:
2193:
2189:
2184:
2182:0-8218-1515-6
2178:
2174:
2167:
2165:
2161:
2157:
2153:
2149:
2144:
2142:
2138:
2131:
2127:
2124:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2099:Bochner space
2097:
2091:
2088:
2086:
2083:
2082:
2078:
2076:
2074:
2071:
2067:
2063:
2059:
2055:
2051:
2047:
2043:
2039:
2035:
2031:
2027:
2025:
2014:
1995:
1970:
1950:
1942:
1926:
1892:
1867:
1840:
1817:
1806:
1798:
1789:
1783:
1760:
1751:
1748:
1725:
1716:
1703:
1700:
1677:
1654:
1627:
1607:
1585:
1581:
1575:
1570:
1567:
1564:
1560:
1556:
1553:
1546:
1542:
1518:
1514:
1507:
1499:
1494:
1491:
1488:
1484:
1477:
1471:
1460:
1435:
1410:
1398:
1382:
1379:
1366:
1363:
1351:
1349:
1347:
1346:
1325:
1316:
1313:
1310:
1299:
1295:
1272:
1268:
1223:
1220:
1213:
1198:
1189:
1186:
1183:
1168:
1160:
1144:
1140:
1094:
1091:
1084:
1083:
1082:
1068:
1048:
1045:
1037:
1021:
999:
995:
991:
985:
979:
959:
956:
948:
905:
902:
899:
884:
882:
880:
876:
860:
851:
848:
837:
833:
829:
825:
821:
816:
803:
800:
777:
761:
754:
752:
738:
735:
732:
722:
716:
712:
701:
698:
695:
691:
686:
681:
665:
653:
652:
649:
628:
625:
622:
612:
608:
584:
575:
562:
559:
551:
547:
526:
522:
515:
505:
502:
499:
495:
491:
486:
480:
476:
465:
462:
459:
455:
450:
445:
370:
367:
364:
354:
350:
339:
335:
319:
310:
297:
291:
285:
282:
276:
270:
267:
261:
258:
255:
249:
205:
185:
178:
177:disjoint sets
162:
149:
146:
138:
134:
118:
115:
108:
82:
69:
68:field of sets
61:
59:
58:values only.
57:
54:
50:
46:
42:
39:defined on a
38:
34:
30:
19:
4367:Main results
4142:
4103:Set function
4031:Metric outer
3986:Decomposable
3843:Cylinder set
3756:
3648:Balanced set
3622:Distribution
3560:Applications
3413:KreinâMilman
3398:Closed graph
3170:Applications
3128:Crinkled arc
3064:PaleyâWiener
2997:
2759:
2743:
2731:
2727:
2724:KluvĂĄnek, I.
2706:
2674:
2666:Bibliography
2638:
2634:
2624:
2596:
2586:
2561:
2557:
2551:
2518:
2514:
2486:
2480:
2462:
2456:
2450:
2446:
2442:
2438:
2434:
2431:
2427:Debreu (1991
2412:
2406:
2380:
2376:
2373:
2343:
2340:Econometrica
2339:
2336:
2298:
2295:Econometrica
2294:
2285:
2273:. Retrieved
2268:
2261:
2260:New Palgrave
2257:
2249:John Roberts
2235:
2200:
2172:
2155:
2151:
2148:KluvĂĄnek, I.
2041:
2022:
2020:
1940:
1717:
1396:
1355:
1343:
1341:
888:
875:real numbers
834:, which are
817:
766:
755:
576:
333:
311:
136:
132:
107:Banach space
65:
32:
26:
4327:compact set
4294:of measures
4230:Pushforward
4223:Projections
4213:Logarithmic
4056:Probability
4046:Pre-measure
3828:Borel space
3746:of measures
3577:Heat kernel
3567:Hardy space
3474:Trace class
3388:HahnâBanach
3350:Topological
2936:Holomorphic
2919:Directional
2879:Derivatives
2515:SIAM Review
2383:(1): 1â12.
2301:(1): 1â17.
873:the set of
336:if for any
53:nonnegative
43:and taking
29:mathematics
4607:Categories
4299:in measure
4026:Maximising
3996:Equivalent
3890:Vitali set
3510:C*-algebra
3325:Properties
2698:0436.28001
2628:Page 210:
2275:7 February
2132:References
2030:non-atomic
2026:'s theorem
1545:partitions
1539:where the
332:is called
4413:Maharam's
4383:Dominated
4196:Intensity
4191:Hausdorff
4098:Saturated
4016:Invariant
3921:Types of
3880:Ï-algebra
3850:đ-system
3816:Borel set
3811:Baire set
3484:Unbounded
3479:Transpose
3437:Operators
3366:Separable
3361:Reflexive
3346:Algebraic
3332:Barrelled
3059:Regulated
3031:Integrals
2750:EMS Press
2447:Convexity
2331:155044347
2060:, and in
2050:economics
2046:zonotopes
1996:μ
1971:μ
1951:μ
1927:μ
1904:Ω
1893:μ
1807:μ
1799:≤
1796:‖
1784:μ
1781:‖
1755:∞
1726:μ
1681:‖
1678:⋅
1675:‖
1561:⋃
1527:‖
1508:μ
1505:‖
1485:∑
1461:μ
1436:μ
1411:μ
1397:variation
1377:→
1364:μ
1221:μ
1173:∞
1092:μ
1069:μ
1022:χ
996:χ
980:μ
855:∞
826:, finite
781:‖
778:⋅
775:‖
707:∞
692:⋃
682:μ
672:∞
669:→
634:∞
585:μ
544:with the
516:μ
511:∞
496:∑
471:∞
456:⋃
446:μ
376:∞
320:μ
286:μ
271:μ
259:∪
250:μ
160:→
147:μ
80:Ω
4430:Fubini's
4420:Egorov's
4388:Monotone
4347:variable
4325:Random:
4276:Strongly
4201:Lebesgue
4186:Harmonic
4176:Gaussian
4161:Counting
4128:Spectral
4123:Singular
4113:s-finite
4108:Ï-finite
3991:Discrete
3966:Complete
3923:Measures
3897:Null set
3785:function
3686:Category
3498:Algebras
3380:Theorems
3337:Complete
3306:Schwartz
3252:glossary
3013:Strongly
2814:Analysis
2255:(eds.).
2227:13064804
2079:See also
2073:analogue
2070:discrete
2024:Lyapunov
1833:for any
1541:supremum
885:Examples
824:measures
729:‖
678:‖
338:sequence
242:one has
66:Given a
37:function
4342:process
4337:measure
4332:element
4271:Bochner
4245:Trivial
4240:Tangent
4218:Product
4076:Regular
4054:)
4041:Perfect
4014:)
3979:)
3971:Content
3961:Complex
3902:Support
3875:-system
3764:Measure
3489:Unitary
3469:Nuclear
3454:Compact
3449:Bounded
3444:Adjoint
3418:Minâmax
3311:Sobolev
3296:Nuclear
3286:Hilbert
3281:Fréchet
3246: (
3179: (
3121:Related
3073:Results
3049:Dunford
3039:Bochner
3005:Bochner
2979:Measure
2655:0514468
2578:1040471
2543:0564562
2535:2029960
2495:0420366
2471:2006785
2421:2525560
2397:0185073
2368:0172689
2360:1913732
2323:0191623
2315:1909854
2219:0920371
1287:-space
1260:to the
1131:to the
1034:is the
972:define
945:of all
137:measure
49:measure
4408:Jordan
4393:Vitali
4352:vector
4281:Weakly
4143:Vector
4118:Signed
4071:Random
4012:Quasi-
4001:Finite
3981:Convex
3941:Banach
3931:Atomic
3759:spaces
3744:
3464:Normal
3301:Orlicz
3291:Hölder
3271:Banach
3260:Spaces
3248:topics
3181:bundle
3009:Weakly
2998:Vector
2772:
2713:
2696:
2686:
2653:
2612:
2576:
2541:
2533:
2493:
2469:
2419:
2395:
2366:
2358:
2329:
2321:
2313:
2225:
2217:
2207:
2179:
2052:, in (
2042:zonoid
2038:convex
2034:closed
1667:Here,
1159:-space
1014:where
830:, and
767:where
546:series
105:and a
45:vector
4250:Young
4171:Euler
4166:Dirac
4138:Tight
4066:Radon
4036:Outer
4006:Inner
3956:Brown
3951:Borel
3946:Besov
3936:Baire
3276:Besov
2902:Total
2531:JSTOR
2467:JSTOR
2417:JSTOR
2356:JSTOR
2327:S2CID
2311:JSTOR
2265:(PDF)
35:is a
4514:For
4403:Hahn
4259:Maps
4181:Haar
4052:Sub-
3806:Atom
3794:Sets
3624:(or
3342:Dual
2770:ISBN
2711:ISBN
2684:ISBN
2610:ISBN
2277:2011
2223:OCLC
2205:ISBN
2177:ISBN
2036:and
1395:the
550:norm
198:and
135:(or
56:real
31:, a
2816:in
2766:114
2694:Zbl
2643:doi
2602:doi
2566:doi
2523:doi
2385:doi
2348:doi
2303:doi
1880:If
1853:in
1640:in
1600:of
1481:sup
1428:of
1038:of
662:lim
218:in
27:In
4609::
3250:â
3011:/
3007:/
2768:.
2748:,
2742:,
2732:20
2692:.
2651:MR
2649:.
2637:.
2608:.
2574:MR
2572:.
2562:28
2560:.
2539:MR
2537:.
2529:.
2519:22
2517:.
2503:^
2491:MR
2445:.
2393:MR
2391:.
2381:12
2379:.
2364:MR
2362:.
2354:.
2344:32
2342:.
2325:.
2319:MR
2317:.
2309:.
2299:34
2297:.
2251:;
2247:;
2221:.
2215:MR
2213:.
2191:^
2163:^
2156:20
2140:^
2056:)
881:.
131:a
4050:(
4010:(
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