Weak Measurability and Characterizations of Risk

Posted: 23 Jun 1999

See all articles by M. Ali Khan

M. Ali Khan

Johns Hopkins University - Department of Economics

Yeneng Sun

National University of Singapore (NUS) - Department of Mathematics; Yale University - Cowles Foundation

Abstract

In the context of a continuum of random variables, arising, for example, as rates of return in financial markets with a continuum of assets, or as individual responses in games with a continuum of players, an important economic issue is to show how idiosyncratic risk can be removed through some device of aggregation or diversification when such risk is explicitly introduced into the model. In this paper, we use recent work of Al-Najjar [1] as a general backdrop to provide a review of the basic issues involved when the continuum is formulated as the Lebesgue interval. We present two examples to argue that the fundamental problem of the non-measurability of sample functions, originally identified by Doob, and further elaborated by Feldman, Gilles and Judd in the economic literature, simply cannot be bypassed by reinterpretations of standard results. We also provide an equivalence result in the spirit of Al-Najjar's efforts; but argue that this elementary result does not go beyond the standard law of large numbers for a sequence of real-valued iid random variables, and as such, is incapable of yielding anything of substantive economic interest beyond this law.

JEL Classification: C60, D80

Suggested Citation

Khan, M. Ali and Sun, Yeneng, Weak Measurability and Characterizations of Risk. Available at SSRN: https://ssrn.com/abstract=161371

M. Ali Khan (Contact Author)

Johns Hopkins University - Department of Economics ( email )

3400 Charles Street
Baltimore, MD 21218-2685
United States

Yeneng Sun

National University of Singapore (NUS) - Department of Mathematics ( email )

Department of Mathematics
Singapore, 117543
Republic of Singapore

Yale University - Cowles Foundation

Box 208281
New Haven, CT 06520-8281
United States

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