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 functionsMathworldPlanetmath fk:M (The index k runs over non-negative integers.) on some measure spaceMathworldPlanetmath M and can find another sequence of measurable functions gk:M such that |fk(x)|gk(x) for all k and almost all x and k=0gk(x) converges for almost all xM and k=0gk(x)𝑑x<. Then


This criterion is a corollary of the monotoneMathworldPlanetmath and dominated convergence theorems. Since the gk’s are nonnegative, the sequence of partial sums is increasing, hence, by the monotone convergence theoremMathworldPlanetmath, Mk=0gk(x)dx<. Since k=0gk(x) converges for almost all 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:


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 gk(x)=1/(x2+k4). We then have -+gk(x)𝑑x=π/k2 and, as k=1k-2<, we can interchange summation and integration:


Doing the integrals, we obtain the answer

Title criterion for interchanging summation and integration
Canonical name CriterionForInterchangingSummationAndIntegration
Date of creation 2013-03-22 16:20:05
Last modified on 2013-03-22 16:20:05
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 9
Author rspuzio (6075)
Entry type Result
Classification msc 28A20