Fubini’s theorem for the Lebesgue integral
In the following suppose we will by convention define in case is not integrable. This simplifies notation and does not affect the results since it will turn out that such cases happen on a set of measure 0.
Also if we have a function then define and .
Suppose and are -finite (http://planetmath.org/SigmaFinite) measure spaces. If then for -almost every and for -almost every . Further the functions and are in and respectively and
You can now see the reason for defining the integral even where and are not integrable since the functions and are normally only almost everywhere defined, and we’d like to define them everywhere. Since we have changed the definition only on a set of measure zero, this does not change the final result and we can interchange the integrals freely without having to worry about where the functions are actually defined.
Note the of this theorem and Tonelli’s theorem for non-negative functions. Here you actually need to check some integrability before switching the integral . A application of Tonelli’s theorem actually shows that you can prove any one of these equations to show that
Theorem (Fubini for sums).
Suppose that is absolutely summable, that is , then
In the above theorem we have used as our set for simplicity and familiarity of notation. Any summable function will have only a countable number of non-zero elements and thus the theorem for arbitrary sets just reduces to the above case.
- 1 Gerald B. Folland. . John Wiley & Sons, Inc., New York, New York, 1999
|Title||Fubini’s theorem for the Lebesgue integral|
|Date of creation||2013-03-22 14:16:09|
|Last modified on||2013-03-22 14:16:09|
|Last modified by||jirka (4157)|