Almost everywhere convergent sequences of weak$^*$-to-norm continuous operators

Let $X$ and $Y$ be Banach spaces, and $T:X^*\to Y$ be an operator. We prove that if $X$ is Asplund and $Y$ has the approximation property, then for each Radon probability $\mu$ on $(B_{X^*},w^*)$ there is a sequence of $w^*$-to-norm continuous operators $T_n:X^*\to Y$ such that $\|T_n(x^*)-T(x^*)\| \to 0$ for $\mu$-a.e. $x^*\in B_{X^*}$; if $Y$ has the $\lambda$-bounded approximation property for some $\lambda\geq 1$, then the sequence can be chosen in such a way that $\|T_n\|\leq \lambda\|T\|$ for all $n\in \mathbb{N}$. The same conclusions hold if $X$ contains no subspace isomorphic to $\ell_1$, $Y$ has the approximation property (resp., $\lambda$-bounded approximation property) and $T$ has separable range. This extends to the non-separable setting a result by Mercourakis and Stamati.