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?

HINT
By Egorov's theorem, convergence a.e. implies for every $\epsilon$ there exists $B$ with $|B| < \epsilon$ such that $f_n\to f$ uniformly on $X\setminus B$ (where $X$ is "almost" the whole space).
Split $\int (f_n - f)g$ in two pieces, one over $B$ and one over $X\setminus B$. On $X\setminus B$ uniform convergence implies the integral can be made as small as you want. Holder's inequality implies the integral on $B$ is controlled by $(M + \|f\|_p)\|g\|_{L^q(B)}$. Taking $B\searrow$ a measure zero set, then the integral on $B$ of $g$ goes to zero. 
Finish by taking a diagonalizing sequence as usual. 
