Weak star convegence in $L^{\infty}(0,T;L^2(\Omega))$ implies almost everywhere convergence?

Question

We know that $u_n \rightarrow^* u$ in $L^{\infty}\left(0, T ; L^2(\Omega)\right)$ to some function $u \in L^{\infty}\left(0, T ; L^2(\Omega)\right)$, i.e. $$ \int_0^T \int_{\Omega} u_n(t, x) v(t, x) d x d t \rightarrow \int_0^T \int_{\Omega} u(t, x) v(t, x) d x d t \quad \text { for all } v \in L^1\left(0, T ; L^2(\Omega)\right) $$

Question: $u_n$ converges to $u$ a.e $(0,T)\times\Omega$? ( or in a subsequence).