By a theorem of Loomis and Sikorski, for every Boolean $\sigma$-algebra $\mathfrak{A}$ there exists a $\sigma$-field of sets $\mathcal{F}$ and a $\sigma$-ideal $\Delta$ such that $\mathfrak{A}$ is isomorphic to $\mathcal{F}/\Delta$.

More precisely, let $X$ be the Stone space of $\mathfrak{A}$, $\mathcal{F}$ be the least $\sigma$-field (of subsets of $X$) containing all open-closed subsets of $X$, and $\Delta$ be the $\sigma$-ideal of all subsets of $\mathcal{F}$ of first category in $X$. Then $\mathfrak{A}$ is isomorphic to $\mathcal{F}/\Delta$.

Does the above theorem hold when $\mathfrak{A}$ is a free Boolean $\sigma$-algebra? In other words, if $X$ denotes the Cantor set, $\mathcal{F}$ the least $\sigma$-field (of subsets of $X$) containing all open-closed subsets of $X$, and $\Delta$ the $\sigma$-ideal of all subsets of $\mathcal{F}$ of first category in $X$, is $\mathfrak{A}$ is isomorphic to $\mathcal{F}/\Delta$.