Let $X$ be a compact metric space, $\mu$ a Borel probability measure on $X$ and $f: X \to \mathbb{R}$ a continuous function. Consider an increasing sequence of $\sigma$-algebras $A_n$ so that for all $n\in \mathbb{N}$, $A_n$ is generated by a finite partition of $X$ into clopen subsets of $X$. Let $A= \sigma(\bigcup_{n\in \mathbb{N}} A_n)$ be the limit $\sigma$-algebra (which is contained in the Borel, as $A_n$ are).

Then it is known that $\mathbb{E}(f\mid A_n) \to \mathbb{E}(f\mid A)$ in the $L^2$ sense. My question is the following: is it true that $\mathbb{E}(f\mid A)$ is $\mu$-almost everywhere equal to a continuous function on $X$?