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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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You should provide some assumption on the sequence $\{h_n\}_n$. –  Siminore Jun 15 '12 at 12:04
    
@Siminore All I know is that $f_n$ is bounded uniformly in the $H^1([0,T]\times \Omega(t))$ norm. –  blahb Jun 15 '12 at 21:26

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