proof of Fatou-Lebesgue theorem


Since |∫g⁒𝑑μ|β‰€βˆ«|g|β’π‘‘ΞΌβ‰€βˆ«Ξ¦β’π‘‘ΞΌ<∞, we have that ∫g⁒𝑑μ>-∞. Similarly, ∫h⁒𝑑μ<∞.

The inequalityMathworldPlanetmath lim infnβ†’βˆžβ‘βˆ«fn⁒𝑑μ≀lim supnβ†’βˆžβ‘βˆ«fn⁒𝑑μ is obvious by definition of lim inf and lim sup.

Define a sequence of functions kn:X→ℝ by kn⁒(x)=fn⁒(x)+Φ⁒(x). Then each kn is nonnegative (since -fn≀|fn|≀Φ) and integrable (since kn≀|fn|+Φ≀2⁒Φ), as is k:=lim infnβ†’βˆžβ‘kn. Fatou’s lemma yields that ∫k⁒𝑑μ≀lim infnβ†’βˆžβ‘βˆ«kn⁒𝑑μ. Thus:

∫g⁒𝑑μ+βˆ«Ξ¦β’π‘‘ΞΌ=∫(g+Ξ¦)⁒𝑑μ=∫k⁒𝑑μ≀lim infnβ†’βˆžβ‘βˆ«kn⁒𝑑μ=lim infnβ†’βˆžβ‘βˆ«(fn+Ξ¦)⁒𝑑μ=lim infnβ†’βˆžβ‘(∫fn⁒𝑑μ+βˆ«Ξ¦β’π‘‘ΞΌ)=lim infnβ†’βˆžβ‘βˆ«fn⁒𝑑μ+lim infnβ†’βˆžβ‘βˆ«Ξ¦β’π‘‘ΞΌ=lim infnβ†’βˆžβ‘βˆ«fn⁒𝑑μ+βˆ«Ξ¦β’π‘‘ΞΌ

Since βˆ«Ξ¦β’π‘‘ΞΌ<∞, it follows that ∫g⁒𝑑μ≀lim infnβ†’βˆžβ‘βˆ«fn⁒𝑑μ.

Note that |-fn|=|fn|≀Φ. Thus,

-∫h⁒𝑑μ =∫-h⁒d⁒μ
=∫-lim supnβ†’βˆžβ‘fn⁒d⁒μ
=∫lim infnβ†’βˆžβ‘(-fn)⁒d⁒μ
≀lim infnβ†’βˆžβ‘βˆ«-fn⁒d⁒μ by a previous ,
=lim infnβ†’βˆžβ‘(-∫fn⁒𝑑μ)
=-lim supnβ†’βˆžβ‘βˆ«fn⁒𝑑μ.

Hence, lim supnβ†’βˆžβ‘βˆ«fnβ’π‘‘ΞΌβ‰€βˆ«h⁒𝑑μ. It follows that -∞<∫g⁒𝑑μ≀lim infnβ†’βˆžβ‘βˆ«fn⁒𝑑μ≀lim supnβ†’βˆžβ‘βˆ«fnβ’π‘‘ΞΌβ‰€βˆ«h⁒𝑑μ<∞. ∎

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