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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.

Suggested Citation

  • Peter J. Hammond & Yeneng Sun, 2000. "Joint Measurability and the One-way Fubini Property for a Continuum of Independent Random Variables," Working Papers 00008, Stanford University, Department of Economics.
  • Handle: RePEc:wop:stanec:00008
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    File URL: http://www-econ.stanford.edu/faculty/workp/swp00008.pdf
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    References listed on IDEAS

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    1. Douglas W. Diamond & Philip H. Dybvig, 2000. "Bank runs, deposit insurance, and liquidity," Quarterly Review, Federal Reserve Bank of Minneapolis, issue Win, pages 14-23.
    2. Celentani, Marco & Pesendorfer, Wolfgang, 1996. "Reputation in Dynamic Games," Journal of Economic Theory, Elsevier, vol. 70(1), pages 109-132, July.
    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. Edward J. Green, 1994. "Individual Level Randomness in a Nonatomic Population," GE, Growth, Math methods 9402001, EconWPA.
    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. Peter J. Hammond & Yeneng Sun, 2003. "Monte Carlo simulation of macroeconomic risk with a continuum of agents: the symmetric case," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 21(2), pages 743-766, March.
    2. Sun, Yeneng & Zhang, Yongchao, 2009. "Individual risk and Lebesgue extension without aggregate uncertainty," Journal of Economic Theory, Elsevier, vol. 144(1), pages 432-443, January.
    3. Peter Hammond & Yeneng Sun, 2008. "Monte Carlo simulation of macroeconomic risk with a continuum of agents: the general case," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 36(2), pages 303-325, August.
    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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