counter-example to Tonelli’s theorem
The following observation demonstrates the necessity of the -finite assumption in Tonelli’s and Fubini’s theorem. Let denote the closed unit interval equipped with Lebesgue measure and the same set, but this time equipped with counting measure . Let
Also observe that there does not exist a simple function on that is dominated by . Hence,
Therefore, the integrand is integrable relative to the product measure. However, as we observed above, the iterated integrals do not agree. This observation illustrates the need for the -finite assumption for Fubini’s theorem.
|Title||counter-example to Tonelli’s theorem|
|Date of creation||2013-03-22 18:16:36|
|Last modified on||2013-03-22 18:16:36|
|Last modified by||rmilson (146)|