Asymptotic arbitrage and the APT with or without measure-theoretic structures

We present a version of the APT based on an asset index set of all arbitrar y infinite cardinality. Under assumptions due to Ross ( 1976. J. Econ Theor y 13, 341-360) and Chamberlain and Rothschild ( 1983. Econometrica 51. 1281 - 1303). we show that, in the absence of gains from asymptotic arbitrage, the square of the deviations of the individual rates of return from a facto r-pricing formula sum to a Finite number and that this absence, while suffi cient, is not necessary for the formula to hold. We relate these results to recent work and explain. in particular. how a version of the APT exhibits several inconsistencies when the index set is the Lebesgue unit interval. ( C) 2001 Academic Press.