Liapounoff's theorem for nonatomic, finitely-additive, bounded, finite-dimensional, vector-valued measures

Authors:
Thomas E. Armstrong and Karel Prikry

Journal:
Trans. Amer. Math. Soc. **266** (1981), 499-514

MSC:
Primary 28B05; Secondary 28A12, 28A60

DOI:
https://doi.org/10.1090/S0002-9947-1981-0617547-8

Erratum:
Trans. Amer. Math. Soc. **272** (1982), 809.

MathSciNet review:
617547

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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, finitely-additive 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 finitely-additive, nonatomic, finite-dimensional, vector-valued measures are fairly arbitrary.

**[1]**Thomas E. Armstrong,*Arrow’s theorem with restricted coalition algebras*, J. Math. Econom.**7**(1980), no. 1, 55–75. MR**568616**, https://doi.org/10.1016/0304-4068(80)90021-X**[2]**Thomas E. Armstrong,*Polyhedrality of infinite dimensional cubes*, Pacific J. Math.**70**(1977), no. 2, 297–307. MR**0493252****[3]**Thomas E. Armstrong and Karel Prikry,*Residual measures*, Illinois J. Math.**22**(1978), no. 1, 64–78. MR**0460581****[4]**Ethan D. Bolker,*A class of convex bodies*, Trans. Amer. Math. Soc.**145**(1969), 323–345. MR**0256265**, https://doi.org/10.1090/S0002-9947-1969-0256265-X**[5]**S. Cobzaş,*Hahn decompositions of finitely additive measures*, Arch. Math. (Basel)**27**(1976), no. 6, 620–621. MR**0425054**, https://doi.org/10.1007/BF01224728**[6]**W. W. Comfort and S. Negrepontis,*The theory of ultrafilters*, Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 211. MR**0396267****[7]**J. Diestel and J. J. Uhl Jr.,*Vector measures*, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR**0453964****[8]**Leonard Gillman and Meyer Jerison,*Rings of continuous functions*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0116199****[9]**E. E. Granirer,*On the range of an invariant mean*, Trans. Amer. Math. Soc.**125**(1966), 384–394. MR**0204551**, https://doi.org/10.1090/S0002-9947-1966-0204551-9**[10]**Paul R. Halmos,*The range of a vector measure*, Bull. Amer. Math. Soc.**54**(1948), 416–421. MR**0024963**, https://doi.org/10.1090/S0002-9904-1948-09020-6**[11]**A. Liapounoff,*Sur les fonctions-vecteurs complètement additives*, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR]**4**(1940), 465–478 (Russian, with French summary). MR**0004080****[12]**A. Liapounoff,*Sur les fonctions-vecteurs complètement additives*, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR]**10**(1946), 277–279 (Russian, with French summary). MR**0017461****[13]**Joram Lindenstrauss,*A short proof of Liapounoff’s convexity theorem*, J. Math. Mech.**15**(1966), 971–972. MR**0207941****[14]**Dorothy Maharam,*Finitely additive measures on the integers*, Sankhyā Ser. A**38**(1976), no. 1, 44–59. MR**0473132****[15]**D. Margolies,*A study of finitely additive measures as regards amenable groups, Liapounov's theorem, and the elimination of infinite integrals via non-standard 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. Ser. A**25**(1978), no. 1, 1–6. MR**0480934****[17]**Leonard J. Savage,*The foundations of statistics*, Second revised edition, Dover Publications, Inc., New York, 1972. MR**0348870****[18]**G. L. Seever,*Measures on 𝐹-spaces*, Trans. Amer. Math. Soc.**133**(1968), 267–280. MR**0226386**, https://doi.org/10.1090/S0002-9947-1968-0226386-5**[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), 839–846. MR**0011164****[21]**A. Sobczyk and P. C. Hammer,*The ranges of additive set functions*, Duke Math. J.**11**(1944), 847–851. MR**0011165****[22]**Eric K. van Douwen and Jan van Mill,*Subspaces of basically disconnected spaces or quotients of countably complete Boolean algebras*, Trans. Amer. Math. Soc.**259**(1980), no. 1, 121–127. MR**561827**, https://doi.org/10.1090/S0002-9947-1980-0561827-0**[23]**Ernst-August Weiss Jr.,*Finitely additive exchange economies*, J. Math. Econom.**8**(1981), no. 3, 221–240. MR**631006**, https://doi.org/10.1016/0304-4068(81)90003-3

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
28B05,
28A12,
28A60

Retrieve articles in all journals with MSC: 28B05, 28A12, 28A60

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1981-0617547-8

Keywords:
Liapounoff's theorem,
finite additivity,
-space,
-algebra,
real-valued measurable cardinal,
uniform measure,
Hahn decomposition,
zonoid

Article copyright:
© Copyright 1981
American Mathematical Society