When $\mu(X)<\infty$, $L^{1}(\mu)\supset L^{2}(\mu)$ and by the Riesz-Fischer theorem, weak convergence of $f_{n}\to f$ in $L^{2}$ is equivalent to $\int_{X}f_{n}\bar{g}\;d\mu\to\int_{X}f\bar{g}\;d\mu$ for each $g\in L^{2}(\mu).$ Taking $g=\chi_{X}$, we have $\int_{X}f_{n}\;d\mu\to\int_{X}f\;d\mu.$ Absent absolute values, this looks like strong (norm) convergence in $L^{1}$. Is there a counter-example to this, and if not, how can it be proved? If the claim is false, can we at least conclude $f_{n_{k}}\to f$ pointwise along some subsequence?
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2Take $X=[0,2\pi]$ and $f_n(t)=e^{int}$. – Etienne Sep 08 '13 at 23:00