Let $U$ be a $C^2-$compact manifold and consider two non negative sequences $f_n,g_n$ such that
$f_n \overset * \to f$ weakly in $L^{\infty}(0,T;\mathcal M_{*}(U))$
$g_n \overset * \to g$ weakly in $\mathcal M([0,T] \times U)$
If $\phi \in C^{\infty}_c([0,T],H^1_0(U))$ then can we deduce anything for the convergence of: $\int_{0}^T (\int_{U} \phi )(\int_{U} {\partial}_t f_n+n{\partial}_tg_n)$ ?
Note: $\mathcal M$ denotes the space of signed measures with finite mass.
MY THOUGHTS:
If instead I had $\int_{0}^T \int_{U} \phi ({\partial}_t f_n+n{\partial}_tg_n)$, I could deduce a convergence in the sense of distributions but now I have no clue how I should handle it.
Any help is much appreciated!
Thanks in advance