proof of Fatou’s lemma

Let f(x)=lim infnfn(x) and let gn(x)=infknfk(x) so that we have


As gn is an increasing sequence of measurable nonnegative functions we can apply the monotone convergence TheoremMathworldPlanetmath to obtain


On the other hand, being gnfn, we conclude by observing

limnXgn𝑑μ=lim infnXgn𝑑μlim infnXfn𝑑μ.
Title proof of Fatou’s lemma
Canonical name ProofOfFatousLemma
Date of creation 2013-03-22 13:29:59
Last modified on 2013-03-22 13:29:59
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 4
Author paolini (1187)
Entry type Proof
Classification msc 28A20