Can we have incomplete markets with a continuum of securities?

Imagine there is a continuum of firms in the economy. Each draws its productivity from the same stochastic process. The stochastic process has unbounded support. The only securities in the economy are the firm's shares.

The claim is that markets are incomplete because the cardinality of the states of the world is $\aleph_2$ and the cardinality of the securities is $\aleph_1$.

However one could also argue that since there is a continuum of firms, the strong law of large numbers applies and one knows exactly the proportion of firms that will get each shock, hence there is no risk (and markets would be complete)

Which of the two statements is correct?

• As a complete layperson of financial economics, I have some clarification questions: It seems to me the first argument assumes generalized continuum hypothesis: $\mathfrak c = \aleph_1$ and $2^{\aleph_1}=\aleph_2$. Why? It seems to me that strong low of large number could only ensure that the sample mean coincide with its expected value. But your second argument is talking about something much stronger, which could be problematic. sciencedirect.com/science/article/pii/0022053185900596 – Metta World Peace Apr 26 '15 at 12:54
• Thanks a lot for the reference. I think the result does hold. Here is what I found: – Daniel Wills Apr 29 '15 at 21:05

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.