A law of large numbers for large economies (*)

Let $X(i),$$i\in [0;1]$ be a collection of identically distributed and pairwise uncorrelated random variables with common finite mean µ and variance $\sigma^{2}.$ This paper shows the law of large numbers, i.e. the fact that $\int^{1}_{0}X(i)di=\mu .$ It does so by interpreting the integral as a Pettis-integral. Studying Riemann sums, the paper first provides a simple proof involving no more than the calculation of variances, and demonstrates, that the measurability problem pointed out by Judd (1985) is avoided by requiring convergence in mean square rather than convergence almost everywhere. We raise the issue of when a random continuum economy is a good abstraction for a large finite economy and give an example in which it is not.