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proof of Fatou-Lebesgue theorem


Since |∫g𝑑μ|β‰€βˆ«|g|π‘‘ΞΌβ‰€βˆ«Ξ¦π‘‘ΞΌ<∞, we have that ∫g𝑑μ>-∞. Similarly, ∫h𝑑μ<∞.

The inequalityMathworldPlanetmath lim infnβ†’βˆžβˆ«fn𝑑μ≀lim supnβ†’βˆžβˆ«fn𝑑μ is obvious by definition of lim inf and lim sup.

Define a sequence of functions kn:X→ℝ by kn(x)=fn(x)+Ξ¦(x). Then each kn is nonnegative (since -fn≀|fn|≀Φ) and integrable (since kn≀|fn|+Φ≀2Ξ¦), as is k:=lim infnβ†’βˆžkn. Fatou’s lemma yields that ∫k𝑑μ≀lim infnβ†’βˆžβˆ«kn𝑑μ. Thus:

∫g𝑑μ+βˆ«Ξ¦π‘‘ΞΌ=∫(g+Ξ¦)𝑑μ=∫k𝑑μ≀lim infnβ†’βˆžβˆ«kn𝑑μ=lim infnβ†’βˆžβˆ«(fn+Ξ¦)𝑑μ=lim infnβ†’βˆž(∫fn𝑑μ+βˆ«Ξ¦π‘‘ΞΌ)=lim infnβ†’βˆžβˆ«fn𝑑μ+lim infnβ†’βˆžβˆ«Ξ¦π‘‘ΞΌ=lim infnβ†’βˆžβˆ«fn𝑑μ+βˆ«Ξ¦π‘‘ΞΌ

Since βˆ«Ξ¦π‘‘ΞΌ<∞, it follows that ∫g𝑑μ≀lim infnβ†’βˆžβˆ«fn𝑑μ.

Note that |-fn|=|fn|≀Φ. Thus,

-∫h𝑑μ =∫-hdΞΌ
=∫-lim supnβ†’βˆžfndΞΌ
=∫lim infnβ†’βˆž(-fn)dΞΌ
≀lim infnβ†’βˆžβˆ«-fndΞΌ by a previous ,
=lim infnβ†’βˆž(-∫fn𝑑μ)
=-lim supnβ†’βˆžβˆ«fn𝑑μ.

Hence, lim supnβ†’βˆžβˆ«fnπ‘‘ΞΌβ‰€βˆ«h𝑑μ. It follows that -∞<∫g𝑑μ≀lim infnβ†’βˆžβˆ«fn𝑑μ≀lim supnβ†’βˆžβˆ«fnπ‘‘ΞΌβ‰€βˆ«h𝑑μ<∞. ∎

Title proof of Fatou-Lebesgue theorem
Canonical name ProofOfFatouLebesgueTheorem
Date of creation 2013-03-22 15:58:50
Last modified on 2013-03-22 15:58:50
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 13
Author Wkbj79 (1863)
Entry type Proof
Classification msc 28A20