counter-example to Tonelli’s theorem

The following observation demonstrates the necessity of the σ-finite assumptionPlanetmathPlanetmath in Tonelli’s and Fubini’s theorem. Let X denote the closed unit interval [0,1] equipped with Lebesgue measureMathworldPlanetmath and Y the same set, but this time equipped with counting measure ν. Let

f(x,y)={1 if x=y,0 otherwise.

Observe that




The iterated integrals do not give the same value, this despite the fact that the integrand is a non-negative function.

Also observe that there does not exist a simple functionMathworldPlanetmathPlanetmath on X×Y that is dominated by f. Hence,


Therefore, the integrand is L1 integrable relative to the product measureMathworldPlanetmath. However, as we observed above, the iterated integrals do not agree. This observation illustrates the need for the σ-finite assumption for Fubini’s theorem.

Title counter-example to Tonelli’s theorem
Canonical name CounterexampleToTonellisTheorem
Date of creation 2013-03-22 18:16:36
Last modified on 2013-03-22 18:16:36
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 4
Author rmilson (146)
Entry type Example
Classification msc 28A35