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

  • Peter J. Hammond
  • Yeneng Sun

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. 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.
  2. Edward J. Green, 1994. "Individual Level Randomness in a Nonatomic Population," GE, Growth, Math methods 9402001, EconWPA.
  3. 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.
  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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