# Weak and Pointwise Convergence question

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|;

$$\bullet$$ $$g_n\rightharpoonup g$$ 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.