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.
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– 2012-03-05