Fatou-Lebesgue theorem

Let (X,μ) be a measure spaceMathworldPlanetmath. If Φ:X is a nonnegative function with Φ𝑑μ<, and if f1,f2, is a sequence of measurable functionsMathworldPlanetmath such that |fn|Φ for each n, then

g=lim infnfnandh=lim supnfn

are both integrable, and

-<g𝑑μlim infnfn𝑑μlim supkfn𝑑μh𝑑μ<.
Title Fatou-Lebesgue theorem
Canonical name FatouLebesgueTheorem
Date of creation 2013-03-22 13:12:53
Last modified on 2013-03-22 13:12:53
Owner Koro (127)
Last modified by Koro (127)
Numerical id 7
Author Koro (127)
Entry type Theorem
Classification msc 28A20
Related topic FatousLemma