proof of dominated convergence theorem

It is not difficult to prove that f is measurable. In fact we can write


and we know that measurable functionsMathworldPlanetmath are closed under the sup and inf operation.

Consider the sequence gn(x)=2Φ(x)-|f(x)-fn(x)|. Clearly gn are nonnegative functions since f-fn2Φ. So, applying Fatou’s Lemma, we obtain

limnX|f-fn|𝑑μlim supnX|f-fn|𝑑μ
= -lim infnX-|f-fn|dμ
= X2Φ𝑑μ-lim infnX2Φ-|f-fn|dμ
X2Φ𝑑μ-X2Φ-lim supn|f-fn|dμ
= X2Φ𝑑μ-X2Φ𝑑μ=0.
Title proof of dominated convergence theoremPlanetmathPlanetmath
Canonical name ProofOfDominatedConvergenceTheorem
Date of creation 2013-03-22 13:30:02
Last modified on 2013-03-22 13:30:02
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 4
Author paolini (1187)
Entry type Proof
Classification msc 28A20
Related topic SecondProofOfDominatedConvergenceTheorem
Related topic SecondProofOfDominatedConvergenceTheorem2