## Joint measurability and the one-way Fubini property for a continuum of independent random variables

HTML articles powered by AMS MathViewer

- by Peter J. Hammond and Yeneng Sun PDF
- Proc. Amer. Math. Soc.
**134**(2006), 737-747 Request permission

## Abstract:

As is well known, a continuous parameter process with mutually independent random variables is not jointly measurable in the usual sense. This paper proposes an extension of the usual product measure-theoretic framework, using a natural “one-way Fubini” property. When the random variables are independent even in a very weak sense, this property guarantees joint measurability and defines a unique measure on a suitable minimal $\sigma$-algebra. However, a further extension to satisfy the usual (two-way) Fubini property, as in the case of Loeb product measures, may not be possible in general. Some applications are also given.## References

- Robert M. Anderson,
*A non-standard representation for Brownian motion and Itô integration*, Israel J. Math.**25**(1976), no. 1-2, 15–46. MR**464380**, DOI 10.1007/BF02756559 - Robert M. Anderson,
*Nonstandard analysis with applications to economics*, Handbook of mathematical economics, Vol. IV, Handbooks in Econom., vol. 1, North-Holland, Amsterdam, 1991, pp. 2145–2208. MR**1207198** - Josef Berger, Horst Osswald, Yeneng Sun, and Jiang-Lun Wu,
*On nonstandard product measure spaces*, Illinois J. Math.**46**(2002), no. 1, 319–330. MR**1936091** - Patrick Billingsley,
*Convergence of probability measures*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0233396** - Donald L. Cohn,
*Measure theory*, Birkhäuser, Boston, Mass., 1980. MR**578344**, DOI 10.1007/978-1-4899-0399-0 - J. L. Doob,
*Stochastic processes depending on a continuous parameter*, Trans. Amer. Math. Soc.**42**(1937), no. 1, 107–140. MR**1501916**, DOI 10.1090/S0002-9947-1937-1501916-1 - J. L. Doob,
*Stochastic processes*, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR**0058896** - Richard Durrett,
*Probability*, The Wadsworth & Brooks/Cole Statistics/Probability Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991. Theory and examples. MR**1068527** - Peter J. Hammond and Yeneng Sun,
*Monte Carlo simulation of macroeconomic risk with a continuum of agents: the symmetric case*, Econom. Theory**21**(2003), no. 2-3, 743–766. Symposium in Honor of Mordecai Kurz (Stanford, CA, 2002). MR**2014352**, DOI 10.1007/s00199-002-0302-y - Kenneth L. Judd,
*The law of large numbers with a continuum of IID random variables*, J. Econom. Theory**35**(1985), no. 1, 19–25. MR**786985**, DOI 10.1016/0022-0531(85)90059-6 - H. Jerome Keisler,
*Hyperfinite model theory*, Logic Colloquium 76 (Oxford, 1976) Studies in Logic and Found. Math., Vol. 87, North-Holland, Amsterdam, 1977, pp. 5–110. MR**0491155** - Peter A. Loeb,
*Conversion from nonstandard to standard measure spaces and applications in probability theory*, Trans. Amer. Math. Soc.**211**(1975), 113–122. MR**390154**, DOI 10.1090/S0002-9947-1975-0390154-8 - Peter A. Loeb,
*A nonstandard functional approach to Fubini’s theorem*, Proc. Amer. Math. Soc.**93**(1985), no. 2, 343–346. MR**770550**, DOI 10.1090/S0002-9939-1985-0770550-9 - Manfred Wolff and Peter A. Loeb (eds.),
*Nonstandard analysis for the working mathematician*, Mathematics and its Applications, vol. 510, Kluwer Academic Publishers, Dordrecht, 2000. MR**1790871**, DOI 10.1007/978-94-011-4168-0 - H. L. Royden,
*Real analysis*, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1963. MR**0151555** - Yeneng Sun,
*Hyperfinite law of large numbers*, Bull. Symbolic Logic**2**(1996), no. 2, 189–198. MR**1396854**, DOI 10.2307/421109 - Yeneng Sun,
*A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN*, J. Math. Econom.**29**(1998), no. 4, 419–503. MR**1627287**, DOI 10.1016/S0304-4068(97)00036-0 - Yeneng Sun,
*The almost equivalence of pairwise and mutual independence and the duality with exchangeability*, Probab. Theory Related Fields**112**(1998), no. 3, 425–456. MR**1660898**, DOI 10.1007/s004400050196 - Sun, Y.N.
*The exact law of large numbers via Fubini extension and the characterization of insurable risks*, J. Econ. Theory, published online, January 2005.

## Additional Information

**Peter J. Hammond**- Affiliation: Department of Economics, Stanford University, 579 Serra Mall, Stanford, California 94305–6072
- Email: peter.hammond@stanford.edu
**Yeneng Sun**- Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543 – and – Department of Economics, National University of Singapore, 1 Arts Link, Singapore 117570
- Email: matsuny@nus.edu.sg
- Received by editor(s): July 31, 2002
- Received by editor(s) in revised form: October 8, 2004
- Published electronically: July 18, 2005
- Additional Notes: Part of this work was done when the first author was visiting Singapore in November 1999 and when the second author was visiting Stanford in July 2002.
- Communicated by: Carl G. Jockusch, Jr.
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**134**(2006), 737-747 - MSC (2000): Primary 28A05, 60G07; Secondary 03E20, 03H05, 28A20
- DOI: https://doi.org/10.1090/S0002-9939-05-08016-0
- MathSciNet review: 2180892