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Infinite product of measurable spaces

Given measure spaces $(X_i, \Sigma_i, \mu_i)$ where $i$ ranges over some arbitrary index set, my understanding is that there has come to be some way to speak of the product of the measure spaces in much the same way that one does a product of 2 measure spaces for Fubini's theorem. Could someone please outline how this is done in the infinite case? I could not find it done in a completely detailed and correct way anywhere. In particular, please state all assumptions like if the measure spaces involved must be $\sigma$-finite. Also, is this construction what is called the "tensor product of measure spaces" or is that a different operation? Please feel free to assume a general knowledge of general analysis at the first year graduate student level or slightly more, but nothing particular about this problem.

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    One can do that meaningfully only for products of probability spaces. The (probably) first rigorous construction of the infinite product of probability spaces can be found in [Notes on infinite product measure spaces, I](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pja/1195573633) by Shizuo Kakutani.2012-10-15

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