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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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    Here is the PlanetMath entry on "infinite product measure," which I believe is what you want: http://planetmath.org/encyclopedia/TotallyFiniteMeasure.html2012-10-13
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    I've seen that, but maybe I've not given it enough thought. First of all, does it really only work on probability spaces? I guess from the perspective of convergence, that kind of makes sense. Secondly, he just lists statements without proof. I suspect he's making extensive use of the caratheodory extension theorem, but the trouble is this is only permitted in the sigma finite case and I don't really know what he's doing.2012-10-13
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    Actually, I've read that and am still looking for the answer to the second part of my question, which is how to define the measure?2012-10-13
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    I would also like to know about the tensor product of measure spaces.2012-10-13
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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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