Tonelli’s theorem

Here denote L+(X) as the space of measurable functionsMathworldPlanetmath X[0,]. Furthermore all integrals are Lebesgue integrals.

Theorem (Tonelli).

Suppose (X,M,μ) and (Y,N,ν) are σ-finite ( measure spacesMathworldPlanetmath. If fL+(X×Y), then the functions xYf(x,y)𝑑ν(y) and yXf(x,y)𝑑μ(x) are in L+(X) and L+(Y) respectively, and furthermore if we denote by μ×ν the product measureMathworldPlanetmath, then


Basically this says that you can switch the of integrals, or integrate over the product space as long as everything is positive and the spaces are σ-finite. Do note that we allow the functions to take on the value of infinityMathworldPlanetmathPlanetmath with the standard conventions used in Lebesgue integration. That is, 0=0, so that if a function is infiniteMathworldPlanetmath on a set of measure 0, then this does not contribute anything to the value of the integral. See the entry on extended real numbers for further discussion.

If we take the counting measure on , then one can the Tonelli theorem for sums.

Theorem (Tonelli for sums).

Suppose that fij0 for all i,jN, then


In the above theorem we have used as our set for simplicity and familiarity of notation. If you would have an uncountable number of non-zero elements fij then all the sums would be infinite and the result would be trivial. So 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 Tonelli’s theorem
Canonical name TonellisTheorem
Date of creation 2013-03-22 14:15:49
Last modified on 2013-03-22 14:15:49
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Theorem
Classification msc 28A35
Related topic FubinisTheorem
Related topic FubinisTheoremForTheLebesgueIntegral