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 (The index runs over
non-negative integers.) on some measure space
and can find another sequence of measurable
functions such that for all and almost all and
converges for almost all and . Then
This criterion is a corollary of the monotone and dominated
convergence theorems. Since the ’s are nonnegative, the
sequence of partial sums is increasing, hence, by the monotone
convergence theorem
, .
Since
converges for almost all ,
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 ’s as simple as possible so that it is easy to integrate them and apply the criterion. A good choice here is . We then have and, as , 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 |