Illinois Journal of Mathematics

Maharam-types and Lyapunov’s theorem for vector measures on Banach spaces

M. Ali Khan and Nobusumi Sagara

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This paper offers a sufficient condition, based on Maharam (Proc. Natl. Acad. Sci. USA 28 (1942) 108–111) and re-emphasized by Hoover and Keisler (Trans. Amer. Math. Soc. 286 (1984) 159–201), for the validity of Lyapunov’s theorem on the range of a nonatomic vector measure taking values in an infinite-dimensional Banach space that is not necessarily separable nor has the Radon–Nikodym property (RNP). In particular, we obtain an extension of a corresponding result due to Uhl (Proc. Amer. Math. Soc. 23 (1969) 158–163). The proposed condition is also shown to be necessary in the sense formalized by Keisler and Sun (Adv. Math. 221 (2009) 1584–1607), and thereby closes a question of long-standing as regards an infinite-dimensional generalization of the theorem. The result is applied to obtain short simple proofs of recent results on the convexity of the integral of a set-valued function, and on the characterization of restricted cores of a saturated economy.

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Illinois J. Math., Volume 57, Number 1 (2013), 145-169.

First available in Project Euclid: 23 June 2014

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Primary: 28B05: Vector-valued set functions, measures and integrals [See also 46G10] 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22] 28B20: Set-valued set functions and measures; integration of set-valued functions; measurable selections [See also 26E25, 54C60, 54C65, 91B14] 46B22: Radon-Nikodým, Kreĭn-Milman and related properties [See also 46G10]


Khan, M. Ali; Sagara, Nobusumi. Maharam-types and Lyapunov’s theorem for vector measures on Banach spaces. Illinois J. Math. 57 (2013), no. 1, 145--169. doi:10.1215/ijm/1403534490.

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  • C. D. Aliprantis and K. C. Border, Infinite dimensional analysis: A hitchhiker's guide, 3rd ed., Springer, Berlin, 2006.
  • N. Azarnia and J. D. M. Wright, On the Lyapunoff–Knowles theorem, Q. J. Math. 33 (1982), 257–261.
  • R. G. Bartle, N. Dunford and J. T. Schwartz, Weak compactness and vector measures, Canad. J. Math. 7 (1955), 289–305.
  • C. Carathéodory, Die homomorphieen von Somen und die Multiplikation von Inhaltsfunionen, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 8 (1939), 8–130.
  • G. Debreu, Integration of correspondences, Proc. fifth Berkeley sympos. math. statist. probability, vol. II: Contributions to probability theory, part 1, Univ. California Press, Berkeley, 1967, pp. 351–372.
  • J. Diestel and J. J. Uhl \bsuffixJr., Vector measures, Amer. Math. Soc., Providence, 1977.
  • N. Dunford and J. T. Schwartz, Linear operators, part I: General theory, Wiley, New York, 1958.
  • A. Dvoretzky, On Liapunov's convexity theorem, Proc. Natl. Acad. Sci. USA 91 (1994), 2145.
  • M. Džamonja and K. Kunen, Properties of the class of measure separable compact spaces, Fund. Math. 147 (1995), 261–277.
  • Ö. Evren and F. Hüsseinov, Theorems on the core of an economy with infinitely many commodities and consumers, J. Math. Econom. 44 (2008), 1180–1196.
  • S. Fajardo and H. J. Keisler, Model theory of stochastic processes, AK Peters, Natick, 2002.
  • D. H. Fremlin, Measure theory, volume 3: Measure algebras, Torres Fremlin, Colchester, 2002.
  • P. R. Halmos and J. von Neumann, Operator methods in classical mechanics, II, Ann. of Math. (2) 43 (1942), 332–350.
  • D. Hoover and H. J. Keisler, Adapted probability distributions, Trans. Amer. Math. Soc. 286 (1984), 159–201.
  • V. M. Kadets, Remark on the Lyapunov theorem on vector measures, Funct. Anal. Appl. 25 (1991), 295–297.
  • V. M. Kadets and M. M. Popov, On the Lyapunov convexity theorem with applications to sign-embeddings, Ukrainian Math. J. 44 (1992), 1091–1098.
  • V. M. Kadets and G. Shekhtman, The Lyapunov theorem for $\ell _p$-valued measures, St. Petersburg Math. J. 4 (1993), 961–966.
  • H. J. Keisler and Y. N. Sun, Why saturated probability spaces are necessary, Adv. Math. 221 (2009), 1584–1607.
  • M. A. Khan, Some remarks on the core of a “large” economy, Econometrica 42 (1974), 633–642.
  • M. A. Khan, Approximately convex average sums of sets in normed spaces, Appl. Math. Comput. 9 (1981), 27–34.
  • M. A. Khan, On the integration of set-valued mappings in a non-reflexive Banach space, II, Simon Stevin 59 (1985), 257–267.
  • M. A. Khan and L. P. Rath, The Shapley–Folkman theorem and the range of a bounded measure: An elementary and unified treatment, Positivity 17 (2013), 381–394.
  • M. A. Khan and N. C. Yannelis, Equilibria in markets with a continuum of agents and commodities, Equilibrium theory in infinite dimensional spaces (M. A. Khan and N. C. Yannelis, eds.), Springer, Berlin, 1991, pp. 233–248.
  • M. A. Khan and Y. C. Zhang, Set-valued functions, Lebesgue extensions and saturated probability spaces, Adv. Math. 229 (2012), 1080–1103.
  • J. F. C. Kingman and A. P. Robertson, On a theorem of Lyapunov, J. London Math. Soc. 43 (1968), 347–351.
  • I. Kluvánek and G. Knowles, Vector measures and control systems, North-Holland, Amsterdam, 1975.
  • G. Knowles, Lyapunov vector measures, SIAM J. Control Optim. 13 (1975), 294–303.
  • H. E. Lacey, The Hamel dimension of any infinite dimensional separable Banach space is $c$, Amer. Math. Monthly 80 (1973), 298.
  • H. E. Lacey, The isometric theory of classical Banach spaces, Springer, Berlin, 1974.
  • J. Lindenstrauss, A short proof of Liapounoff's convexity theorem, J. Math. Mech. 15 (1966), 971–972.
  • P. A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113–122.
  • P. A. Loeb and Y. N. Sun, Purification and saturation, Proc. Amer. Math. Soc. 137 (2009), 2719–2724.
  • A. Liapounoff, Sur les fonctions-vecteurs complètement additives, Bull. Acad. Sci. URSS. Sér. Math. 4 (1940), 465–478 (in Russian).
  • D. Maharam, On homogeneous measure algebras, Proc. Natl. Acad. Sci. USA 28 (1942), 108–111.
  • K. Podczeck, On the convexity and compactness of the integral of a Banach space valued correspondence, J. Math. Econom. 44 (2008), 836–852.
  • H. P. Rosenthal, On the injective Banach spaces and the spaces $L^\infty(\mu)$ for finite measures $\mu$, Acta Math. 124 (1970), 205–248.
  • H. L. Royden, Real analysis, 3rd ed., Macmillan, New York, 1988.
  • W. Rudin, Functional analysis, McGraw-Hill, New York, 1973.
  • A. Rustichini and N. C. Yannelis, What is perfect competition? Equilibrium theory in infinite dimensional spaces (M. A. Khan and N. C. Yannelis, eds.), Springer, Berlin, 1991, pp. 249–265.
  • E. Saab and P. Saab, Lyapunov convexity type theorems for non-atomic vector measures, Quaest. Math. 26 (2003), 371–383.
  • D. Schmeidler, A remark on the core of an atomless economy, Econometrica 40 (1972), 579–580.
  • Y. N. Sun, On the theory of vector valued Loeb measures and integration, J. Funct. Anal. 104 (1992), 327–362.
  • Y. N. Sun, Distributional properties of correspondences on Loeb spaces, J. Funct. Anal. 139 (1996), 68–93.
  • Y. N. Sun, Integration of correspondences on Loeb spaces, Trans. Amer. Math. Soc. 349 (1997), 129–153.
  • Y. N. Sun and N. C. Yannelis, Saturation and the integration of Banach valued correspondences, J. Math. Econom. 44 (2008), 861–865.
  • J. J. Uhl \bsuffixJr., The range of a vector-valued measure, Proc. Amer. Math. Soc. 23 (1969), 158–163.
  • R. G. Ventner, Liapounoff convexity-type theorems, Vector measures, integration and related topics (G. P. Curbera, G. Mockenhaupt and W. J. Ricker, eds.), Birkhäuser, Berlin, 2010, pp. 371–380.