criterion for interchanging summation and integration

The following criterion for interchanging integration and summationis often useful in practise: Suppose one has a sequence of measurablefunctions $f_k:M\to ℝ$ (The index $k$ runs overnon-negative integers.) on some measure space $M$and can find another sequence of measurablefunctions $g_k:M\to ℝ$ such that $|f_k(x)|\le g_k(x)$ for all $k$ and almost all $x$ and ${\sum }_{k=0}^{\mathrm{\infty }}{g}_k(x)$converges for almost all $x\in M$ and . Then

$${\int }_M\sum _{k=0}^{\mathrm{\infty }}{f}_k(x)dx=\sum _{k=0}^{\mathrm{\infty }}{\int }_M{f}_k(x)𝑑x$$

This criterion is a corollary of the monotone and dominatedconvergence theorems. Since the $g_k$’s are nonnegative, thesequence of partial sums is increasing, hence, by the monotoneconvergence theorem, .Since${\sum }_{k=0}^{\mathrm{\infty }}{g}_k(x)$ converges for almost all $x$,

$$\left|\sum _{k=0}^n{f}_k(x)\right|\le \sum _{k=0}^n|f_k(x)|\le \sum _{k=0}^n{g}_k(x)\le \sum _{k=0}^{\mathrm{\infty }}{g}_k(x),$$

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

As an example of this method, consider the following:

$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\sum _{k=1}^{\mathrm{\infty }}\frac{\cos (x/k)}{x^2+k^4}dx$$

The idea behind the method is to pick our $g$’s as simple as possibleso that it is easy to integrate them and apply the criterion. A goodchoice here is $g_k(x)=1/(x^2+k^4)$. We then have ${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{g}_k(x)𝑑x=\pi /k^2$ and, as, we can interchange summationand integration:

$$\sum _{k=1}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{\cos (x/k)}{x^2+k^4}𝑑x.$$

Doing the integrals, we obtain the answer

$$\pi \sum _{k=1}^{\mathrm{\infty }}\frac{e^{-k}}{k^2}$$