Since a σ-algebra in measure theory is indeed an algebra over $\mathbb{Z}_2$ with addition given by symmetric difference and multiplication given by intersection, does it mean we can put measure on any $\mathbb{Z}_2$ algebra (aka Boolean ring)? In particular, it would mean that we can define integration on any idempotent ring with all results from measure theory holding. Am I missing anything, and has this been explored?
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1$\begingroup$ What would you be trying to integrate? One integrates functions (which are not elements of the $\sigma$-algebra), not subsets. You could maybe build some formal theory (consider the completion of the group algebra of your ring in some appropriate topology …), but the Stone representation theorem says that any algebra is an algebra of subsets anyway, so it seems like you'd be quite close to doing measure theory while carefully avoiding naming the set. $\endgroup$– LSpiceCommented Jul 1, 2020 at 22:41
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2$\begingroup$ Do you want your measure theory to be $\sigma$-additive? $\endgroup$– Michael GreineckerCommented Jul 1, 2020 at 22:48
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2$\begingroup$ You wrote "$\mathbf{Z}_2$-algebra (aka Boolean ring)" but obviously not every $\mathbf{Z}_2$-algebra is a Boolean ring. $\endgroup$– YCorCommented Jul 1, 2020 at 22:55
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$\begingroup$ @LSpice You can identify a measurable function with the function that maps Borel sets to their preimages. This gives you a ring homomorphism from the Borel sets to the ring of measurable sets and you can build a theory of integration for such homomorphisms. $\endgroup$– Michael GreineckerCommented Jul 1, 2020 at 23:03
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1$\begingroup$ See John M. H. Olmsted, Lebesgue theory on a Boolean algebra, Transactions of the American Mathematical Society 51 #1 (January 1942), pp.164-193 AND Roman Sikorski, The integral in a Boolean algebra, Colloquium Mathematicum 2 #1 (1949), pp. 20-26. $\endgroup$– Dave L RenfroCommented Jul 2, 2020 at 6:53
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1 Answer
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According to Proposition 416Q(b) in Fremlin's Measure Theory, finitely additive functionals A→[0,∞) are in a canonical bijective correspondence with finite Radon measures on the Stone space Spec(A) of A, which is a compact Hausdorff totally disconnected topological space.
This means that we can integrate any continuous function Spec(A)→R with respect to a finitely additive functional on A. This type of integrand is very general, and, in particular, includes all continuous homomorphisms from the Boolean algebra of Borel subsets of R to the Boolean algebra A, as alluded to in the comments.
answered Jul 2, 2020 at 1:05
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$\begingroup$ Fremlin's book can be found here: www1.essex.ac.uk/maths/people/fremlin/mt.htm $\endgroup$ Commented Jul 2, 2020 at 7:09