# Products of infinitely many measure spaces. Applications?

• What are some typical applications of the theory for measures on infinite product spaces?
• Are there any applications that you think are particularly interesting - that make the study of this worthwhile beyond finite products, Fubini-Tonelli.
• Are there theorems that require, or are equivalent to, certain choice principles (AC, PIT, etc)? (similar to Tychonoff in topology)

Sorry for being so vague, I am just trying to get a feel for this new area before diving head-first into the technical details.

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Infinite products of measure spaces are used very frequently in probability. Probabilists are frequently interested in what happens asymptotically as a random process continues indefinitely. The Strong Law of Large Numbers, for example, tells us that if $\{X_i\}_i$ is a sequence of independent, identically distributed random variables with finite mean $\mu$ then the sum $\frac{1}{n}\sum_{i=1}^n X_i$ converges almost surely to $\mu$. But how do we find infinitely many independent random variables to which we can apply this theorem? The most common way to produce these variables is with the infinite product. For example, say we want to flip a coin infinitely many times. A way to model this would be to let $\Omega$ be the probability space $\{-1,1\}$ where $P(1) = P(-1) = \frac{1}{2}$. Then we consider the probability space $\prod_{i=1}^\infty \Omega$, and let $X_i$ be the $i$th component. Then the $X_i$ are independent identically distributed variables.