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Let $\{(X_ \lambda, \mathcal{A} _\lambda, \mu _\lambda)\} _ {\lambda \in \Lambda}$ be an arbitrary collection of probability measure spaces. (Meaning each $\mu_ \lambda(A)=1$.)

Is there a natural way to construct a probability measure from these on $X=\prod_{ \lambda \in \Lambda} X_ \lambda$?

Of course, one could define something really trivial and stupid like $(X,\{ \emptyset,X\}, \mu)$ where $\mu( \emptyset)=0$ and $\mu(X)=1$. But I want to know if we can do better.

I was thinking of trying to define a measure on $\mathcal{A}= \{ \prod_{ \lambda \in \Lambda} A_ \lambda: A_\lambda \in \mathcal{A}_ \lambda\}$, but I'm not even sure this is a $\sigma$-algebra anyways.

William Elliot
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Pascal's Wager
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1 Answers1

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The natural way to define the $\sigma$-algebra on product space have a very similar objective like the construction of the topology on cartesian product of topologic spaces. The idea is construct the coarset $\sigma$-algebra that’s make every projection measurable functions, you $\textbf{collect}$ the inverse image of measurable sets by all single projections $$\pi_\beta : \prod_\lambda X_\lambda\to X_\beta,\,\,\,\,\beta\in\Lambda$$ and consider the coarset $\sigma$-algebra that contais the above $\textbf{collection}$. In the link given by @saulspatz good details are explained about the above $\sigma$-algebra.

There is many ways to construct the required probability measure on the product space with the previous $\sigma$-algebra. In pratice the most important result is named Kolmogorov Extension Theorem and have many aplications on Stochastic theory, the result requires some compatibility condition to works.

Stochastic Student
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