<p>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.</p>

<p>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.</p>

CC BY-SA 3.0
2015-04-29T21:20:18.610Z
2015-04-29T21:20:18.610Z
Daniel Wills
<p>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. </p>

<p>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 $\aleph_1$.</p>

<p>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)</p>

<p>Which of the two statements is correct?</p>

CC BY-SA 3.0
2015-04-26T12:54:19.710Z
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. http://www.sciencedirect.com/science/article/pii/0022053185900596
Metta World Peace
CC BY-SA 3.0
2015-04-29T21:05:44.260Z
Thanks a lot for the reference. I think the result does hold. Here is what I found:
Daniel Wills
<portfolio-theory><complete-markets>
Can we have incomplete markets with a continuum of securities?
Daniel Wills
# Can we have incomplete markets with a continuum of securities?

(Asked by: username_0 on 2015-04-25)

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?

**Tags:** portfolio-theory, complete-markets

> **Comments:**
> **username_1**: 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. http://www.sciencedirect.com/science/article/pii/0022053185900596  
> **username_0**: Thanks a lot for the reference. I think the result does hold. Here is what I found:  

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## 1 answers

### Answer by username_0 on 2015-04-29. Score: 1 | Accepted answer

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.


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