Fubini’s theorem

Fubini’s theorem Let IN and JM be compact intervals, and let f:I×JK be a Riemann integrablePlanetmathPlanetmath function such that, for each xI the integral


exists. Then F:IK is Riemann integrable, and


This theorem effectively states that, given a function of N variables, you may integrate it one variable at a time, and that the order of integration does not affect the result.

Example Let I:=[0,π/2]×[0,π/2], and let f:I,xsin(x)cos(y) be a function. Then


Note that it is often simpler (and no less correct) to write ∫⋯∫If as If.

Title Fubini’s theorem
Canonical name FubinisTheorem
Date of creation 2013-03-22 13:39:13
Last modified on 2013-03-22 13:39:13
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 11
Author mathcam (2727)
Entry type Theorem
Classification msc 26B12
Related topic TonellisTheorem
Related topic FubinisTheoremForTheLebesgueIntegral
Related topic IntegrationUnderIntegralSign