Weakly convergence and pointwise convergence of $L^2$ How  to show the below theorem? In fact ,I feel it is not right , if $f(x)\ne 0 $ at zero measure set  , I still have 
$$
\int_\Omega f(x)\varphi(x)dx = 0 ~~~\forall \varphi \in C^\infty_0(\Omega)
$$
What is my mistake ? 




Since we work in $L^2$-space, $f=0$ is understood as $f$ belongs to the equivalence class of the function identically equal to $0$ for the equivalence relation "equal almost everywhere".
