For example, say $\Omega$ is a finite set and $(\Omega,S,\mu)$ is a probability triple defining the uniform distribution over atoms of $\Omega$.
The meet algebra of two sub-algebras of $S$ is well defined by intersection and a join algebra can be generated from the union. Do these join and meet operations on the collection of sub-algebras of S themselves form a Boolean algebra?
I am going down the list of axioms and can readily establish several of them. I am not sure if both Distributivity laws hold. I am really looking for a reference or counterexample for this seemingly basic question.
Each finite Boolean algebra is isomorphic to the powerset $\mathcal{P}(X)$ of a finite set $X$. The Boolean subalgebras of $\mathcal{P}(X)$ ordered by inclusion form a lattice which is isomorphic to the order dual of the lattice of equivalence relations on $X$.
(Proof sketch: given an equivalence relation $\theta$ on $X$, the subsets of $X$ closed under $\theta$ form a Boolean subalgebra. Conversely, given a Boolean subalgebra $\mathbf{B}$ of $\mathcal{P}(X)$, consider the equivalence relation $\theta$ such that $\langle a, b \rangle \in \theta$ if and only if for each $U \in \mathbf{B}$ we have $a \in U \iff b \in U$.)
This lattice is not distributive for $| X | \geq 3$.