What's wrong with this argument?
Let $f_n$ be a sequence of functions such that $f_n \to f$ in $L^2(\Omega)$. This means $\lVert f_n - f \rVert_{L^2(\Omega)} \to 0,$ i.e., $\int_\Omega(f_n - f)^2 \to 0.$ Since the integrand is positive, this must mean that $f_n \to f$ a.e.
Why is this not true? Apparently this only true for a subsequence $f_n$ (and in all $L^p$ spaces).