Let $\Omega$ be a bounded and smooth domain. Suppose I have a sequence of non-negative functions $u_n \in L^\infty((0,1)\times \Omega) \cap L^\infty((0,1);L^\infty(\Omega))$ with $$0 \leq u_n \leq 1 \quad \text{a.e}$$ and $$u_n \rightharpoonup^* u \quad\text{in $L^\infty((0,1)\times \Omega)$}$$ to some $u$.

Is it possible to conclude that for a subsequence, $u_{n_j}(t) \rightharpoonup^* u(t)$ in $L^\infty(\Omega)$ for a.e. $t$? The subsequence $\{n_j\}$ should not depend on the point $t$.

Additional info:

For each $n$, $u_n$ is a piecewise constant function, i.e., $u_n = \sum_{k=1}^n a_{kn}\chi_{I_{kn}}(t)$ holds for a partition $\{I_{kn}\}$ and $a_{kn} \in L^\infty(\Omega)$.

- The sequence $u_n(t)$ is bounded uniformly in $L^\infty(\Omega)$ for a.e. $t$.