Liapounoff's theorem for nonatomic, finitelyadditive, bounded, finitedimensional, vectorvalued measures
Authors:
Thomas E. Armstrong and Karel Prikry
Journal:
Trans. Amer. Math. Soc. 266 (1981), 499514
MSC:
Primary 28B05; Secondary 28A12, 28A60
Erratum:
Trans. Amer. Math. Soc. 272 (1982), 809.
MathSciNet review:
617547
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Abstract: Liapounoff's theorem states that if is a measurable space and is nonatomic, bounded, and countably additive, then is compact and convex. When is replaced by a complete Boolean algebra or an algebra (to be defined) and is allowed to be only finitely additive, is still convex. If is any Boolean algebra supporting nontrivial, nonatomic, finitelyadditive measures and is a zonoid, there exists a nonatomic measure on with range dense in . A wide variety of pathology is examined which indicates that ranges of finitelyadditive, nonatomic, finitedimensional, vectorvalued measures are fairly arbitrary.
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 [1]
 T. E. Armstrong, Arrow's theorem with restricted coalition algebras, J. Math. Econom. 7 (1980), 5575. MR 568616 (81h:90010)
 [2]
 , Polyhedrality of infinite dimensional compact cubes, Pacific J. Math. 70 (1977), 297307. MR 0493252 (58:12281)
 [3]
 T. E. Armstrong and K. Prikry, Residual measures, Illinois J. Math. 22 (1978), 6478. MR 0460581 (57:574)
 [4]
 E. D. Bolker, A class of convex bodies, Trans. Amer. Math. Soc. 145 (1969), 323345. MR 0256265 (41:921)
 [5]
 S. Cobzas, Hahn decompositions of finitely additive measures, Arch. Math. (Basel) 27 (1976), 620621. MR 0425054 (54:13012)
 [6]
 W. Comfort and S. Negrepontis, The theory of ultrafilters, Springer, New York, 1974. MR 0396267 (53:135)
 [7]
 J. Diestel and J. J. Uhl, Jr., Vector measures, Math. Surveys, no. 15, Amer. Math. Soc., Providence, R. I., 1977. MR 0453964 (56:12216)
 [8]
 L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, Princeton, N. J., 1960. MR 0116199 (22:6994)
 [9]
 E. E. Granirer, On the range of an invariant mean, Trans. Amer. Math. Soc. 125 (1966), 384394. MR 0204551 (34:4390)
 [10]
 P. R. Halmos, The range of a vector measure, Bull. Amer. Math. Soc. 54 (1948), 416421. MR 0024963 (9:574h)
 [11]
 A. Liapounoff, Sur les fonctionsvecteurs completement additives, Izv. Akad. Nauk SSSR 4 (1940), 465478. MR 0004080 (2:315e)
 [12]
 , Sur les fonctionsvecteurs completement additives, Izv. Akad. Nauk SSSR 10 (1946), 277279. MR 0017461 (8:157b)
 [13]
 J. Lindenstrauss, A short proof of Liapounoff's convexity theorem, J. Math. Mech. 15 (1966), 971972. MR 0207941 (34:7754)
 [14]
 D. Maharam, Finitely additive measures on the integers, Sankhyā Ser. A 38 (1976), 4449. MR 0473132 (57:12810)
 [15]
 D. Margolies, A study of finitely additive measures as regards amenable groups, Liapounov's theorem, and the elimination of infinite integrals via nonstandard real numbers, Dissertation, Univ. of California, Berkeley, Calif., 1978.
 [16]
 K. P. S. Bhaskara Rao and M. Bhaskara Rao, Existence of nonatomic charges, J. Austral. Math. Soc. A 25 (1978), 16. MR 0480934 (58:1081)
 [17]
 L. J. Savage, The foundations of statistics, Dover, New York, 1972. MR 0348870 (50:1364)
 [18]
 G. L. Seever, Measures on spaces, Trans. Amer. Math. Soc. 133 (1968), 267280. MR 0226386 (37:1976)
 [19]
 Z. Semandeni, Banach spaces of continuous functions. I, PWN, Warsaw, 1971.
 [20]
 A. Sobczyk and P. C. Hammer, A decomposition of additive set functions, Duke Math. J. 11 (1944), 839846. MR 0011164 (6:129d)
 [21]
 , The ranges of additive set functions, Duke Math. J. 11 (1944), 847851. MR 0011165 (6:129e)
 [22]
 E. K. van Douwen and J. van Mill, Subspaces of basically disconnected spaces or quotients of countably complete Boolean algebras, Trans. Amer. Math. Soc. 259 (1980), 121127. MR 561827 (81b:54038)
 [23]
 E. A. Weiss, Finitely additive exchange economies, J. Math. Econom. (to appear). MR 631006 (83a:90038)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198106175478
PII:
S 00029947(1981)06175478
Keywords:
Liapounoff's theorem,
finite additivity,
space,
algebra,
realvalued measurable cardinal,
uniform measure,
Hahn decomposition,
zonoid
Article copyright:
© Copyright 1981
American Mathematical Society
