Let $A$ be the free $\sigma$-algebra on $\omega$ free generators and $X$ its Stone space. Then $A$ is isomorphic to the quotient algebra $Ba(X)/M$, where $Ba(X)$ is the $\sigma$-field of Baire subsets of $X$, and $M$ is the $\sigma$-ideal of meagre Baire sets.
What does $M$ contain?
I am asking the question because $A$ is atomic and $\sigma$-distributive, and I found a theorem in Sikorski's book Boolean Algebras (theorem 24.5, p. 99) that states that such $\sigma$-algebras are isomorphic to a $\sigma$-field of sets. This would mean that $M$ is empty, which I find hard to believe... Any clarification is welcome.
I don't see the logic in why that fact would imply $M$ is empty: $\sigma$-fields of sets aren't always empty, are they?
A trivial example of a member of $M$ is any $\{x\}$, where $x \in X$ and is non-isolated; and also countable unions of such singletons. Also most non-trivial compact Hausdorff $X$ will contain Cantor sets (with empty interior), which are also example member sets.