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Joint Measurability and the One-way Fubini Property for a Continuum of Independent Random Variables

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  • Peter J. Hammond
  • Yeneng Sun

Abstract

April 2000 As is well known, a continuous parameter process with mutually independent random variables is not jointly measurable in the usual sense. This paper proposes using a natural ``one-way Fubini'' property that guarantees a unique meaningful solution to this joint measurability problem when the random variables are independent even in a very weak sense. In particular, if F is the smallest extension of the usual product sigma-algebra such that the process is measurable, then there is a unique probability measure v on F such that the integral of any v-integrable function is equal to a double integral evaluated in one particular order. Moreover, in general this measure cannot be further extended to satisfy a two-way Fubini property. However, the extended framework with the one-way Fubini property not only shares many desirable features previously demonstrated under the stronger two-way Fubini property, but also leads to a new characterization of the most basic probabilistic concept --- stochastic independence in terms of regular conditional distributions.

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Bibliographic Info

Paper provided by Stanford University, Department of Economics in its series Working Papers with number 00008.

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Date of creation: Apr 2000
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Handle: RePEc:wop:stanec:00008

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  1. Edward J. Green, 1994. "Individual Level Randomness in a Nonatomic Population," GE, Growth, Math methods 9402001, EconWPA.
  2. Diamond, Douglas W & Dybvig, Philip H, 1983. "Bank Runs, Deposit Insurance, and Liquidity," Journal of Political Economy, University of Chicago Press, vol. 91(3), pages 401-19, June.
  3. Anderson, Robert M., 1991. "Non-standard analysis with applications to economics," Handbook of Mathematical Economics, in: W. Hildenbrand & H. Sonnenschein (ed.), Handbook of Mathematical Economics, edition 1, volume 4, chapter 39, pages 2145-2208 Elsevier.
  4. Celentani, Marco & Pesendorfer, Wolfgang, 1996. "Reputation in Dynamic Games," Journal of Economic Theory, Elsevier, vol. 70(1), pages 109-132, July.
  5. Lucas, Robert Jr. & Prescott, Edward C., 1974. "Equilibrium search and unemployment," Journal of Economic Theory, Elsevier, vol. 7(2), pages 188-209, February.
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Cited by:
  1. Sun, Yeneng & Zhang, Yongchao, 2008. "Individual Risk and Lebesgue Extension without Aggregate Uncertainty," MPRA Paper 7448, University Library of Munich, Germany.
  2. Peter Hammond & Yeneng Sun, 2008. "Monte Carlo simulation of macroeconomic risk with a continuum of agents: the general case," Economic Theory, Springer, vol. 36(2), pages 303-325, August.
  3. Peter J. Hammond & Yeneng Sun, 2003. "Monte Carlo simulation of macroeconomic risk with a continuum of agents: the symmetric case," Economic Theory, Springer, vol. 21(2), pages 743-766, 03.
  4. Hammond, Peter J. & Sun, Yeneng, 2007. "Monte Carlo Simulation of Macroeconomic Risk with a Continuum Agents : The General Case," The Warwick Economics Research Paper Series (TWERPS) 803, University of Warwick, Department of Economics.

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