These are really two questions, but the second presupposes the first.


First, let $( B\_i )\_{i\in I}$ be an arbitrary family of Boolean algebras. I want to directly form a product of them that is like the product topology on the product of their Stone spaces, ideally without using AC (or BPI, pun not intended). I haven't worked with Boolean algebras as such before--no doubt this will show in what I say--and my little literature search finds at least one author doing products of Boolean algebras by passing to Stone spaces, which I don't want to do. 


I am thinking something along the lines of the space of finite formal joins of finite formal meets. If I can't find anything, I guess what I'll do will be something like define the "cylinder elements" as finite sets $\{(i\_1,a\_1),...,(i\_n,a\_n)\}$ of index-element pairs, with $a\_j \in B\_{i\_j}$ and the $i\_j$ all distinct, and then take the space of finite sets of cylinder elements, modulo an appropriate equivalence relation. But I don't want to reinvent the wheel or multiply nonstandard notation. I'd much rather be able to cite something and use whatever notation is standard. A good feature from my point of view of the above partially-described construction is that if $I\subset J$, then the product of $(B\_i)\_{i\in I}$ is a subalgebra of the product of $(B\_i)\_{i\in J}$--I'd ideally like a construction that does that.


The second thing is that I need a result showing that if I have a (finitely-additive) probability measure $P\_i$ on each $B\_i$ (satisfying the obvious Boolean algebra analogues of the finitely-additive probability axioms), then I can get the natural measure on the product measure. (In the above notation, we will want $P(\{\{(i\_1,a\_1),...,(i\_n,a\_n)\}\}) = \Pi\_{j=1}^n P\_{i\_j}(a\_{i\_j})$.) Again, this can't be hard to prove, but I don't want to spend space writing up a proof of this if I can just find a citation.



Products of Boolean algebras and probability measures thereon
This is called the "direct sum" or "internal sum" of Boolean algebras. See *Introduction to Boolean Algebras* by Givant and Halmos, p. 427 for the abstract definition and p. 432 for the concrete description you want (the elements of the internal sum are finite joins of finite meets of elements and complements of elements from the disjoint union of the summands).


