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OK, so now I have $\int_0^T{\phi(t)h_n(t)} \to \int_0^T{\phi(t)h(t)}$ which holds for all $\phi \in C^\infty_c(0,T).$ From this how can I deduce that $h_n \to h$ (a.e)?

I think I need maybe something to do with Fatou but I am not sure.

($h_n = \frac{d}{dt}\int_{\Omega(t)}{f_ng}$, and $h$ is similar but without the $n$. Also I know that $f_n$ converges to $f$ in $L^2(O,T;L^2(\Omega(t))$.)

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    @Siminore All I know is that $f_n$ is bounded uniformly in the $H^1([0,T]\times \Omega(t))$ norm.2012-06-15

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