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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

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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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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. [Downloadable!]
  2. Celentani, Marco & Pesendorfer, Wolfgang, 1996. "Reputation in Dynamic Games," Journal of Economic Theory, Elsevier, vol. 70(1), pages 109-132, July. [Downloadable!] (restricted)
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  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. [Downloadable!] (restricted)
  4. Lucas, Robert Jr. & Prescott, Edward C., 1974. "Equilibrium search and unemployment," Journal of Economic Theory, Elsevier, vol. 7(2), pages 188-209, February. [Downloadable!] (restricted)
  5. 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. [Downloadable!]
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(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Peter Hammond & Yeneng Sun, 2001. "Monte Carlo Simulation of Macroeconomic Risk with a Continuum of Agents: The Symmetric Case," Working Papers 01015, Stanford University, Department of Economics. [Downloadable!]
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  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. [Downloadable!] (restricted)
  3. 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. [Downloadable!]
  4. Sun, Yeneng & Zhang , Yongchao, 2008. "Individual Risk and Lebesgue Extension without Aggregate Uncertainty," MPRA Paper 7448, University Library of Munich, Germany. [Downloadable!]
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