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I'm looking for a reference for the existence of a finitely-additive product probability measure for an arbitrary family of finitely-additive probability measures. It's easy to prove, but I'd like to have a citation rather than to include a notationally messy proof. If I was looking for the harder countably-additive version, I guess I'd cite Kakutani (the result is credited in that case to Lomnicki-Ulam, but I've heard that they have a mistake in the proof).

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On what sets should the product probability be defined? –  Michael Greinecker Feb 24 '13 at 0:23
And what should a product measure satisfy? Fubini's theorem does not hold for finitely additive probabilities. –  Michael Greinecker Feb 24 '13 at 2:09
I'd like the product probability to be defined on the field generated by products of subsets of the respective spaces almost all of which are improper, assigning to such a set the product of the respective measures of the proper subsets. That said, I am not sure I need an answer to the question any more. This came up in the context of probability measures on Boolean algebras, and I was going to move to the Stone spaces and use probability measures on them. But now I want to work directly with the Boolean algebras to avoid or minimize AC use. –  Alexander Pruss Feb 24 '13 at 14:16
Most proofs of the Daniell-Kolmogorov consistency theorem consist of two steps: Showing that a finitely additve measure on the field is induced. Using a regularity argument to show it is sigma-additive. So, there should be lots of proofs you could cite. –  Michael Greinecker Feb 24 '13 at 14:34
Right, but the proofs will generally assume that the measures we start with are sigma-additive. The first step of the proof won't actually use the sigma-additivity, and so I can cite the first step by saying "by the first step of the proof of the Ulam-Lomnicki/Daniell-Kolmogorov in [reference], which step assumes but does not actually use sigma-additivity". But that makes a nuisance for the referee--if I were the referee, I might feel obliged to go look up the reference to make sure sigma-additivity isn't used in the proof. Still, it may be moot as now I want to work with BAs directly. –  Alexander Pruss Feb 24 '13 at 14:52

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