Let $(g_{n})_{n}$ be bounded and measurable where $g_{n}\xrightarrow{n \to \infty} g, \mu-$a.e. and $f_{n} \xrightarrow{L^{p}}f$. I need to show that $g_{n}f_{n} \xrightarrow{L^{1}}gf$, and my proof is fine until I need to show that:
$\int_{0}^{1}|f|^{p}|g_{n}-g|^pd\mu\xrightarrow{n \to \infty}0$
Any ideas on how I can show this?
Since $g_{n}\xrightarrow{n \to \infty} g, \mu-$a.e., it follows that $|g_{n}|^{p}\xrightarrow{n \to \infty} |g|^p, \mu-$a.e. it is also true that $g_{n} \leq M$ for all $n \in \mathbb N$ and therefore $g_{n}\xrightarrow{L^{p}}g$. Stuck on how to exclude the $|f|^{p}$ from $\int_{0}^{1}|f|^{p}|g_{n}-g|^pd\mu$