Proving that:$\int_X f_n g \, d\mu \to \int_X fg \, d\mu$ for all $g$ in $\mathscr{L}^q (X)$

Solution 1:

Your answer is correct but since $\mu(X)<\infty$ But that is a typical consequence of Vitali's convergence Theorem. Here is the generalization.

Theorem Assume $\mu(X)<\infty$ and $1<p<\infty$.

Let $(u_n)_n$ be a bounded sequence of $L^p(X)$ converging almost everywhere to $u$. Then $(u_n)_n$ converges to $u$ in $L^r(X)$ for all $1\leq r<p$.

Proof

The $r$-tightness $(u_n)_n$ obviously holds true since $\mu(X)<\infty$.

The $r$-uniform integrability follows from Hölder inequality.

\begin{align*} \sup_{n\geq 1} \int_E |u_n(x)|^r\,d \mu(x)&\leq \mu(E)^{1-\frac{r}{p}} \sup_{n\geq 1} \big(\int_X |u_n(x)|^p\,d \mu(x)\big)^{r/p}\\ &\leq C \mu(E)^{1-\frac{r}{p}}\to0. \end{align*} as $\mu(E)\to 0$.