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# criterion for interchanging summation and integration

The following criterion for interchanging integration and summation is often useful in practise: Suppose one has a sequence of measurable functions $f_{k}\colon M\to\mathbb{R}$ (The index $k$ runs over non-negative integers.) on some measure space $M$ and can find another sequence of measurable functions $g_{k}\colon M\to\mathbb{R}$ such that $|f_{k}(x)|\leq g_{k}(x)$ for all $k$ and almost all $x$ and $\sum_{{k=0}}^{\infty}g_{k}(x)$ converges for almost all $x\in M$ and $\sum_{{k=0}}^{\infty}\int g_{k}(x)\,dx<\infty$. Then

$\int_{M}\sum_{{k=0}}^{\infty}f_{k}(x)\,dx=\sum_{{k=0}}^{\infty}\int_{M}f_{k}(x% )\,dx$ |

This criterion is a corollary of the monotone and dominated convergence theorems. Since the $g_{k}$’s are nonnegative, the sequence of partial sums is increasing, hence, by the monotone convergence theorem, $\int_{M}\sum_{{k=0}}^{\infty}g_{k}(x)\,dx<\infty$. Since $\sum_{{k=0}}^{\infty}g_{k}(x)$ converges for almost all $x$,

$\left|\sum_{{k=0}}^{n}f_{k}(x)\right|\leq\sum_{{k=0}}^{n}|f_{k}(x)|\leq\sum_{{% k=0}}^{n}g_{k}(x)\leq\sum_{{k=0}}^{\infty}g_{k}(x),$ |

the dominated convergence theorem implies that we may integrate the sequence of partial sums term-by-term, which is tantamount to saying that we may switch integration and summation.

As an example of this method, consider the following:

$\int_{{-\infty}}^{{+\infty}}\sum_{{k=1}}^{\infty}{\cos(x/k)\over x^{2}+k^{4}}% \,dx$ |

The idea behind the method is to pick our $g$’s as simple as possible so that it is easy to integrate them and apply the criterion. A good choice here is $g_{k}(x)=1/(x^{2}+k^{4})$. We then have $\int_{{-\infty}}^{{+\infty}}g_{k}(x)\,dx=\pi/k^{2}$ and, as $\sum_{{k=1}}^{\infty}k^{{-2}}<\infty$, we can interchange summation and integration:

$\sum_{{k=1}}^{\infty}\int_{{-\infty}}^{{+\infty}}{\cos(x/k)\over x^{2}+k^{4}}% \,dx.$ |

Doing the integrals, we obtain the answer

$\pi\sum_{{k=1}}^{\infty}{e^{{-k}}\over k^{2}}$ |

## Mathematics Subject Classification

28A20*no label found*

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