proof of Fatou-Lebesgue theorem
Since |β«gπΞΌ|β€β«|g|πΞΌβ€β«Ξ¦πΞΌ<β, we have that β«gπΞΌ>-β. Similarly, β«hπΞΌ<β.
The inequality lim infnβββ«fnπΞΌβ€lim supnβββ«fnπΞΌ is obvious by definition of lim inf and lim sup.
Define a sequence of functions kn:Xββ by kn(x)=fn(x)+Ξ¦(x). Then each kn is nonnegative (since -fnβ€|fn|β€Ξ¦) and integrable (since knβ€|fn|+Ξ¦β€2Ξ¦), as is k:=lim infnββkn. Fatouβs lemma yields that β«kπΞΌβ€lim infnβββ«knπΞΌ. Thus:
β«gπΞΌ+β«Ξ¦πΞΌ=β«(g+Ξ¦)πΞΌ=β«kπΞΌβ€lim infnβββ«knπΞΌ=lim infnβββ«(fn+Ξ¦)πΞΌ=lim infnββ(β«fnπΞΌ+β«Ξ¦πΞΌ)=lim infnβββ«fnπΞΌ+lim infnβββ«Ξ¦πΞΌ=lim infnβββ«fnπΞΌ+β«Ξ¦πΞΌ
Since β«Ξ¦πΞΌ<β, it follows that β«gπΞΌβ€lim infnβββ«fnπΞΌ.
Note that |-fn|=|fn|β€Ξ¦. Thus,
-β«hπΞΌ | =β«-hdΞΌ |
=β«-lim supnββfndΞΌ | |
=β«lim infnββ(-fn)dΞΌ | |
β€lim infnβββ«-fndΞΌ by a previous , | |
=lim infnββ(-β«fnπΞΌ) | |
=-lim supnβββ«fnπΞΌ. |
Hence, lim supnβββ«fnπΞΌβ€β«hπΞΌ. It follows that -β<β«gπΞΌβ€lim infnβββ«fnπΞΌβ€lim supnβββ«fnπΞΌβ€β«hπΞΌ<β. β
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 |