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.