Real Integral Series Convergence
Why
Sums of non-negative functions are increasing, and workable with the monotone convergence theorem.
Result
The integral of the limit of the partial sums
of a sequence of measurable, nonnegative,
extended-real-valued functions is the limit of
the partial sums of the integrals.
Let $(X, \mathcal{A}, \mu )$ be a measure
space, and let $\seqt{f}: \to \nneri$ a
$\mathcal{A}$-measurable function for every
natural number $n$.
We want to show that:
\[ \int \sum_{k = 1}^{\infty} f_k d\mu = \sum_{k = 1}^{\infty} \int f_k d\mu. \]
We apply the monotone convergence theorem to the sequence $\set{\sum_{i = 1}^{n}f_i}_n$. This sequence is nondecreasing because $f_n \geq 0$ for all $n$.