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If $f_n \in L_2(\mu)$, $f_n\rightarrow f$ almost everywhere, this is not enough to conclude $f\in L_1(\mu)$.

But is it enough to conclude whether $f\in L_2(\mu)$ or $$\lim_{n \to \infty}\int_{R}{|f_n(x)-f(x)|}^2<\infty$$

What about the assumption change to $\sup\int_{R}{|f_n(x)|}^2d(\mu)<\infty$

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    If you change the assumption to $$\sup_n \int_{\mathbb{R}} |f_n(x)|^2 dx < \infty$$ then you can at least conclude that $f\in L^2$ by Fatou's lemma.2012-10-27

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