Weak convergence in $L^p$ space 
Let $(\Omega,\mathfrak{B},\mu)$ be a measure space and  $1<p<\infty$. Consider a sequence $f_n\in L^p(\Omega,\mathfrak{B},\mu))$ s.t $f_n\to f$ a.e and $\operatorname{Sup}\{\|f_n\|_p:n\in\Bbb{N}\}=C<\infty$. Prove that for every $\phi\in (L^p(\Omega,\mathfrak{B},\mu))^*, \phi(f_n)\to\phi(f)$.

My attempt: By Riesz representation theorem, every $\phi\in (L^p(\Omega,\mathfrak{B},\mu))^*$ corresponds to a $g\in L^q(\Omega,\mathfrak{B},\mu)$. So let $g\in  L^q(\Omega,\mathfrak{B},\mu)$, I want to prove that $\int_{\Omega}f_ngd\mu\to\int_{\Omega}fgd\mu$.
My attempt: It's easy to show, using Fatou's lemma, that $\|f\|_p\leq C$. Moreover, if we assume that $\mu$ is finite, then using Egoroffs theorem proves the claim.
Now I want to use similar arguments to the case $\mu$ is not finite. Since $\|g\|_q<\infty$, I know that the sets $E_N=\{\omega:|g(\omega)|>\frac{1}{N}\}$ are of finite measure. So I define $E=\bigcup_{N\in\Bbb{N}}E_N$. It suffices to show that $\int_{E}f_ngd\mu\to\int_{E}f_gd\mu$. I know this is true for every $E_N$, but I wasn't able to conclude it for $E$. Any help would be appreciated.
**edit:
I realize this has been asked several times, but not in this generality and I wasn't able to follow the solutions suggested or verify them
 A: In order to prove that $\int_{\Omega}f_ngd\mu\to\int_{\Omega}fgd\mu$, it suffices to deal with the case where $g$ takes the form  $\mathbf{1}_{A_i}$, where $A\in \mathfrak B$ and has finite measure. Indeed, let $N\in\mathbb N$ and $c_1,\dots,c_N\in\mathbb R$, $A_1,\dots,A_N\in\mathfrak B$. From
$$
\left\lvert \int_{\Omega}f_ngd\mu-\int_{\Omega}fgd\mu\right\rvert
\leqslant \left\lvert \int_{\Omega}f_n\left(\sum_{i=1}^Nc_i\mathbf{1}_{A_i}\right)d\mu-\int_{\Omega}f\left(\sum_{i=1}^Nc_i\mathbf{1}_{A_i}\right)d\mu\right\rvert+\left\lvert \int_{\Omega}f_n\left(g-\sum_{i=1}^Nc_i\mathbf{1}_{A_i}\right) d\mu\right\rvert +\left\lvert \int_{\Omega}f\left(g-\sum_{i=1}^Nc_i\mathbf{1}_{A_i}\right) d\mu\right\rvert 
\\
\leqslant \left\lvert \int_{\Omega}f_n\left(\sum_{i=1}^Nc_i\mathbf{1}_{A_i}\right)d\mu-\int_{\Omega}f\left(\sum_{i=1}^Nc_i\mathbf{1}_{A_i}\right)d\mu\right\rvert+2C\left\lVert g-\sum_{i=1}^Nc_i\mathbf{1}_{A_i}\right\rVert_q,
$$
we would  get that
$$
\limsup_{n\to\infty}\left\lvert \int_{\Omega}f_ngd\mu-\int_{\Omega}fgd\mu\right\rvert 
\leqslant 2C\left\lVert g-\sum_{i=1}^Nc_i\mathbf{1}_{A_i}\right\rVert_q
$$
and the fact the class consisting of functions of the form $\sum_{i=1}^Nc_i\mathbf{1}_{A_i}$ is dense in  $\mathbb L^q$ would give the wanted convergence.
You are thus reduced to the case where the measure is finite.
