I'm trying to prove the following exercise:
Suppose that $f,f_n,g\in\mathcal{L}^2(\mathbb{R})$, $n=1,2,...$, $f_n$ converges to $f$ $\mu-$almost everywhere, and $$ \int_{\mathbb{R}}|f_n(x)|^2 dx\leq 1 , n=1,2,... $$ then we have: $$ \lim_{n\rightarrow \infty}\int_{\mathbb{R}}|f_n(x)-f(x)||g(x)|dx=0 $$ Here are my attempts:
use the Cauchy inequality, we have: $$ \left(\int_{\mathbb{R}}|f_n(x)-f(x)||g(x)|dx\right)^2\leq \int_{\mathbb{R}}|f_n(x)-f(x)|^2dx\int_{\mathbb{R}}|g(x)|^2dx $$ and since $g\in \mathcal{L}^2(\mathbb{R})$, $\int_{\mathbb{R}}|g(x)|^2dx<\infty$, it suffices to prove that: $$ \lim_{n\rightarrow \infty}\int_{\mathbb{R}}|f_n(x)-f(x)|^2dx=0 $$ but I was stucked there since I can't find a dominating function and apply the Lebesgue Dominated Convergence Theorem. I also tried Vitali Convergence Theorem, but in vain as well. Can anybody give me some hints?