Can we have incomplete markets with a continuum of securities?
Thanks a lot for the reference. I think the result does hold. Here is what I found:
The validity of such a law of large numbers what subject to some debate in the 1980s.
See Judd (1985), Feldman and Gilles (1985) and Uhlig (1996) for representative papers.

Luckily Feldman and Gilles show that for a probability space $(Y; B(Y ); \Pi)$ there exists a continuum of random variables $y(i;.) : \Omega \rightarrow Y$ such that for all $i$ the random variable is distributed according to $\Pi$ and that for all $\omega \in \Omega$ and all $D \in b(Y)$
$$m(i \in I : \{y(i; \omega) \in D ) = \Pi(D)$$
that is, the population income distribution is nonstochastic and given by $\Pi$ The continuum of random variables cannot be pairwise independent.