I want to show the following:
Let $(X,\mathcal{A}, \mu)$ be a measurable space. Consider $L^p(X,\mathcal{A}, \mu)$ where $1 <p < \infty$ and $q$ is the conjugate of $p$. Further let $(f_n)_n$ be a sequence in $L^p(X,\mathcal{A}, \mu)$, that converges to $f \in L^p(X,\mathcal{A}, \mu)$. Show that for every $g \in L^q(X,\mathcal{A}, \mu)$
$\int_X f_ng d\mu \rightarrow \int_X fg d\mu$.
My attempt: Since I have to show $\int_X f_ng \rightarrow \int_X fg$, let's consider
$|\int_X f_ng d\mu - \int_X fg d\mu|$
Thinking about how to get an estimate, I tought about the Hölder Inequality. Which states that for measurable $f$ and $g$ on $X$ such that $|f|< \infty$ and $|g| \infty$ almost everywhere on $X$ and $\frac{1}{p}+\frac{1}{q}=1$ we have $||fg||_1 \leq ||f||_p||g||_q$ Thus I get $|\int_X f_ng d\mu - \int_X fg d\mu|\leq \int_X|f_ng-fg| d \mu=\int_X|g||f_n-f| d \mu$. Since $f_n,f \in L^p$, $L^p$ a vector space $\Rightarrow f_n-f \in L^p$. How applying the Hölder Inequality yields $\int_X |g| |f_n-f| d \mu \leq ||g||_q ||f_n-f||_p$. Now, since $||f_n-f||_p \rightarrow 0$, we get the result.
I have two questions. Is my solution correct? And does this hold for $p=1$ and $q=\infty$ too?