Convergence of integrals in $L^p$ Stuck with this problem from Zgymund's book. 
Suppose that $f_{n} \rightarrow f$ almost everywhere and that $f_{n}, f \in L^{p}$ where $1<p<\infty$. Assume that $\|f_{n}\|_{p} \leq M < \infty$. Prove that:
$\int f_{n}g \rightarrow \int fg$ as $n \rightarrow \infty$ for all $g \in L^{q}$ such that $\dfrac{1}{p} + \dfrac{1}{q} = 1$.
Right, so I estimate the difference of the integrals and using Hölder end up with:
$$\left|\int f_{n} g - \int fg\right| \leq \|g\|_{q} \|f_{n} - f\|_{p}$$
From here I'm stuck because we are not assuming convergence in the seminorm but just pointwise convergence almost everywhere. How to proceed?

Here is a proof which is not based on Egoroff's theorem. As Jonas T points out in an other answer, Fatou's Lemma imply
$$\int|f|^p=\int\lim |f_n|^p =\int\liminf |f_n|^p  \le \liminf\int |f_n|^p \le M^p$$
and hence $f\in L^p$. 
At this point we may assume $f=0$ (consider the problem with $f_n$ replaced by $f_n-f$), and we must show $\int f_ng\to 0$.
Fix a number $A\gt0$, and let $E_n=\{x: |f_n(x) g(x)|\le A|g(x)|^q\}$ where $q=p/(p-1)$ (the conjugate to $p$). By dominated convergence we have
$$\int_{E_n}f_ng\,dx\to0$$
On the complement, $E_n^c =X-E_n$, we have $|g|^{q}\le A^{-1}|f_n g|$. Hence, by Hölder's inequality
$$\int_{E_n^c}|f_ng|\,dx\le\left(\int_{E_n^c} |f_n|^p\right)^{1/p}\left(\int_{E_n^c} |g|^q\right)^{1/q}\le M A^{-1/q}\left(\int_{E_n^c}|f_ng|\,dx\right)^{1/q}$$
In other words
$$\left(\int_{E_n^c}|f_ng|\,dx\right)^{1-1/q}\le MA^{-1/q}$$
that is 
$$\limsup_n\int_{E_n^c}|f_ng|\,dx\le M^pA^{-p/q}.$$
In total we have shown 
$$\limsup_n\int|f_ng|\,dx\le M^pA^{-p/q}$$
for all $A\gt0$, from which be conclude $$\lim_n\int|f_ng|=0.$$
