Almost everywhere pointwise convergence vs pointwise convergence

Question

Let $X$ and $Y$ be compact metric spaces. Suppose that $\{f^n\}$ is a sequence of continuous maps $f^n:X\to Y$, and suppose that for every Borel probability measure $\mu$ on $(X,\mathcal B(X))$, there exists $f_\mu:X\to Y$ such that some subsequence of $f^n$ converges pointwise to $f_\mu$ on a set $X^*\in\mathcal B(X)$ with $\mu(X^*)=1$. Can we conclude that there exist $f:X\to Y$ and a subsequence $\{f^{n_k}\}$ such that $f^{n_k}\to f$ pointwise?