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Assume that $f,f_j\subset L^2(\mathbb{R}^n)$ for $j=1,2,3,\dots$, $f_j\to f$ a.e and $\int f_j^2\to \int f^2 dx$.

Prove $\int|f_j-f|^2\to 0$.

Robert Z
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user144600
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    This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level. –  Sep 03 '16 at 17:43
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    A particular case of http://math.stackexchange.com/questions/51502/if-f-k-to-f-a-e-and-the-lp-norms-converge-then-f-k-to-f-in-lp – Robert Z Sep 03 '16 at 17:44

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Since $$ |f_j-f|^2 \leq 4(|f_j|^2 + |f|^2) $$ you can apply Fatou's lemma to the difference $$ g_j = 4(|f_j|^2 + |f|^2) - |f_j-f|^2 $$ which is non-negative and converges to $8|f|^2$. This gives you what you want.

Prahlad Vaidyanathan
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