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I found an interesting property in some lecture notes on weak convergence: lemma 3.2 (2) on page 10: https://www.uio.no/studier/emner/matnat/math/MAT4380/v06/Weakconvergence.pdf . The idea is as follows:

Let $(X,\mu)$ be a $\sigma$ finite measure space and $(f_n)_{n\in\mathbb{N}^*}, (g_n)_{n\in\mathbb{N}^*}, f,g :X\to \overline{\mathbb{R}}$ are $\mu$ measurable functions. Prove that if :

$\bullet$ There is a function $f:X\to\overline{\mathbb{R}}$ such that $\lim\limits_{n\to\infty} f_n(x)= f(x),$ for $\mu$ - almost all $x\in X$. (pointwise convergence);

$\bullet$ There is some positive constant $M>0$ such that: $||f_n||_{\infty}:=\mathrm{ess sup}\ |f_n|<M, \forall n\in\mathbb{N}^*$;

$\bullet$ $g_n\rightharpoonup g$ in $L^1(X,\mu)$,

then $f_n g_n\rightharpoonup fg$ in $L^1(X,\mu)$ i.e. $\lim\limits_{n\to\infty} \displaystyle\int_{X} f_n g_n h\ d\mu=\int_{X} fgh\ d\mu, \forall h\in L^{\infty}(X,\mu)$

I tried to prove this by definition but I cannot make good estimates on $\int_{X} |g_n||f_n-f|\ d\mu$. In the lecture notes sais that this is a simple property and the proof is omitted...Any help is very welcomed.

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