# Compute limit of integral

Let ${f_{n}}$ be a sequence in $L^{2}(X,\mu)$ such that $||f_{n}||_{2} \rightarrow 0$ as $n \rightarrow \infty$. How to show that:

$\displaystyle \lim_{n \to \infty} \int_{X} |f_n(x)| \log(1+|f_{n}(x)|) d\mu = 0$

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Use the fact that $\log (1+y) \leq y$.
So we get $\displaystyle \int_{X} |f_n(x)| \log(1+|f_n(x)|) d \mu \leq \displaystyle \int_{X} |f_n(x)| |f_n(x)| d \mu = \displaystyle \int_{X} |f_n(x)|^2 d \mu = ||f_n||_2^2$.
Also, note that $\displaystyle \int_{X} |f_n(x)| \log(1+|f_n(x)|) d \mu \geq 0$.
Hence, $0 \leq \displaystyle \lim_{n \rightarrow \infty} \displaystyle \int_{X} |f_n(x)| \log(1+|f_n(x)|) d \mu \leq \lim_{n \rightarrow \infty} ||f_n||_2^2 = 0$
@user10: Yes. If you have a measurable function $f$ and another measurable function $g$, then $g(f)$ is also measurable. Hence LHS is measurable and then you can argue as I said. –  user17762 Nov 19 '10 at 23:18