Fubini’s theorem for the Lebesgue integral

This is the version of the Fubini’s Theorem for the Lebesgue integralMathworldPlanetmath. For the Riemann integral, see the standard calculus version (http://planetmath.org/FubinisTheorem).

In the following suppose we will by convention define Xf𝑑μ:=0 in case f 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 measureMathworldPlanetmath 0.

Also if we have a function f:X×Y𝔽 then define fx(y):=f(x,y) and fy(x):=f(x,y).

Theorem (Fubini).

Suppose (X,M,μ) and (Y,N,ν) are σ-finite (http://planetmath.org/SigmaFinite) measure spaces. If fL1(X×Y) then fxL1(ν) for μ-almost every x and fyL1(μ) for ν-almost every y. Further the functions xYfx𝑑ν and yXfy𝑑ν are in L1(μ) and L1(ν) respectively and


You can now see the reason for defining the integral even where fx and fy are not integrable since the functions xYfx𝑑ν and yXfy𝑑ν 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 zeroMathworldPlanetmath, 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 fL1(X×Y)

X[Y|f(x,y)|𝑑ν(y)]𝑑μ(x)<, or 

If we take the counting measure on , then one can the Fubini theoremPlanetmathPlanetmath for sums.

Theorem (Fubini for sums).

Suppose that fij is absolutely summable, that is i,jN|fij|<, then


In the above theorem we have used as our set for simplicity and familiarity of notation. Any summable function fij 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
Canonical name FubinisTheoremForTheLebesgueIntegral
Date of creation 2013-03-22 14:16:09
Last modified on 2013-03-22 14:16:09
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 9
Author jirka (4157)
Entry type Theorem
Classification msc 28A35
Synonym Fubini’s theorem
Related topic FubinisTheorem
Related topic TonellisTheorem