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 an arbitrary infinite cardinality. Under assumptions due to Ross (1976, J. Econ. Theory 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 factor-pricing formula sum to a finite number and that this absence, while sufficient, 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. Journal of Economic Literature Classification Numbers: G12, C60.