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Let $\{(X_i,\Sigma_i, \mu_i)\}_{i=1}^\infty$ be a collection of $\sigma$-finite measure space.
For each $i$, let $f_i$ be a $\mu_i$-measurable nonnegative function.
Define the product measure space $ \{X,\Sigma,\mu\}=\{\prod_{i=1}^\infty X_i, \prod_{i=1}^\infty \Sigma_i, \prod_{i=1}^\infty \mu_i \} $ and the function $g: X \to \mathbb R_+$ $ g(x)=\prod_{i=1}^\infty f_i(x_i). $

Is Tonelli theorem applicable to $ \int_X g(x) d\mu(x) \; ? $

That is, order of integration can be interchanged at will to get $ \int_X g(x) d\mu(x) = \prod_{i=1}^\infty \int_{X_i} f(x_i)d\mu_i(x). $

I would say yes, by inductively applying Tonelli theorem pairwise, then taking the limit.
But I'm not sure it's correct.

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    It seems that in general the function $g$ is not measurable with respect to the product sigma-algebra $\otimes_i\Sigma_i$, a fact which would make pointless the rest of the question.2012-10-26

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