Q:

$L^p$ convergences implies to interchange integral and sum

Suppose we have function $f_n$ all p-integrable for $p<\infty$. We define new measurable functions $g_n$ as:
$$g_n := \sum_{i=1}^n f_i$$
Further we assume that $g_n\to g$ in $L^p$. Now why is the following true ($\mu# probability measure):
$$ \int{g d\mu} = \int{\lim_{n\to \infty}\sum_{i=1}^n f_id\mu} = \lim_{n\to \infty} \sum_{i=1}^n \int{f_id\mu}$$
The question is, why could we interchange the limit and the integral? If we know that $f_n\ge 0$ this is clear. But what is for example if $f_n$ are random variables? 
Thanks for your help.

A:

Call $X$ the measure space, and let $q\ge1$ be such that $1/p+1/q=1$. By Hölder's inequality
$$\begin{align*}
\Bigl|\int_X(g_n-g)d\mu\Bigr|&\le\int_X|g_n-g|\,d\mu\\
&\le\Bigl(\int_xd\mu\Bigr)^{1/q}\Bigl(\int_X|g_n-g|^pd\mu\Bigr)^{1/p}\\
&=\|g-g_n\|_{L^p(X)}\to0.
\end{align*}$$

