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?



Measure theory on abstract Boolean ring
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.


