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.