Examples of Complete Atomless Boolean Algebras

Question

What are some natural examples of complete atomless Boolean algebras? I am aware of the regular open algebra on the Euclidean space. But are there more? Thanks!

Answer 1

Well, the regular open algebra $RO(X)$ of any topological space $X$ is always a complete Boolean algebra. If $X$ is Hausdorff then $RO(X)$ is atomless iff $X$ has no isolated points. So this gives lots of examples, and in fact every example is of this form, since every complete Boolean algebra is the regular open algebra of some Hausdorff space (namely its Stone space). (Many of these examples will be isomorphic to each other; for instance, if $X$ is a nonempty separable metric space without isolated points then $RO(X)\cong RO(\mathbb{R})$. But you can easily find lots of non-isomorphic examples by taking large spaces for $X$ such that $RO(X)$ will have different cardinality, for instance.)

Some other naturally occuring examples come from measure theory. If $(X,A,\mu)$ is a $\sigma$-finite measure space and $N\subseteq A$ is the ideal of null sets, then the quotient algebra $A/N$ is complete, and is atomless iff the measure $\mu$ is atomless. So for instance, the algebra of Lebesgue measurable subsets of $\mathbb{R}$ modulo null sets is a very natural example of an atomless complete Boolean algebra, which is famously not isomorphic to $RO(\mathbb{R})$ (for instance, you can prove that $RO(\mathbb{R})$ admits no nontrivial $\sigma$-additive measure, by an argument similar to the construction of fat Cantor sets).