Let $X$ be a compact Hausdorff space.
I know that every Borel set $B$ is congruent to a regular closed set $R$ modulo a meager set $M$. In other words, $B\oplus R=M$ (where $\oplus$ denotes the operation of symmetric difference).
I would like to show that this decomposition is unique in the sense that there exists a unique $R$ such that $B\oplus R=M$. I know this involves the Baire category theorem. How does the argument unfold?
If $G \oplus P = H \oplus Q$ where $G,H$ are regular open, then check that
$H \setminus \overline{G} \subseteq H \oplus G = P \oplus Q$ and so $H \setminus \overline{G}$ is an open set which is meagre, so empty ($X$ is a Baire space), so $H \subseteq \overline{G}$, so $H \subseteq G$ as both sets are regular open. By symmetry $G = H$ and we have unicity.