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A strong version of Dominated Convergence Theorem - Real Analysis

  1. Nov 24, 2013 #1
    1. The problem statement, all variables and given/known data

    Let [itex](g_{n})_{n \in \mathbb{N}}[/itex] a sequence functions integrable over [itex]\mathbb{R}^{p}[/itex] such that:
    [itex]g_{n} (x) \longrightarrow g(x)[/itex] almost everywhere in [itex]\mathbb{R}^{p}[/itex], where [itex]g[/itex] is a function integrable over [itex]\mathbb{R}^{p}[/itex].
    Given [itex](f_{n})_{n \in \mathbb{N}}[/itex] a sequence of functions measurable over [itex]\mathbb{R}^{p}[/itex] such that:
    [itex]| f_{n} | \leq g_{n}[/itex], for all [itex]n \in \mathbb{N}[/itex] and [itex]f_{n}(x) \longrightarrow f(x)[/itex] a.e in [itex]\mathbb{R}^{p}[/itex], prove that:
    [itex]\displaystyle \int_{R^{p}} g(x) dx = \lim_{n \to \infty} \displaystyle \int_{\mathbb{R}^{p}} g_{n} (x) dx \Longrightarrow \displaystyle \int_{R^{p}} f(x) dx = \lim_{n \to \infty} \displaystyle \int_{\mathbb{R}^{p}} f_{n} (x) dx [/itex]


    2. The attempt at a solution

    I tried to [itex]g_{n} - | f_{n} |[/itex] as a sequence of nonnegative functions, but I need the sequence to be an increasing sequence to be able to apply Beppo-Levi's theorem.

    Fatou's lemma didn't help me neither:

    [itex] \displaystyle \int g-|f|\leq \liminf_{n \to \infty} \displaystyle \int g_{n} - |f_{n}| [/itex]
    [itex] \displaystyle \int - | f | \leq \liminf_{n \to \infty} \displaystyle \int - | f_{n} | [/itex]
    [itex] \limsup_{n \to \infty} \displaystyle \int |f_{n}| \leq \displaystyle \int |f |[/itex]
    And by Fatou's lemma:
    [itex] \displaystyle \int | f | \leq \liminf_{n \to \infty} \displaystyle \int |f_{n}| [/itex]
    As [itex] \liminf \leq \limsup [/itex] both limits are the same and
    [itex] \lim_{n \to \infty} \displaystyle \int |f_{n}| = \displaystyle \int |f | [/itex]

    Instead of using [itex]| f |[/itex] I could use [itex] f^{+} [/itex] and [itex] f^{-} [/itex] and prove the theorem?

    I couldn't solve this problem in weeks, I was going to post it to get some help and I only tried to do it once more because I don't like to have to read my handwrite to transcribe it to the PC and I wanted to show that I really tried but it appears that I finally did it, it's not the first time that the exercise of writing it in this forum made me solve a problem, but this time I'm posting it anyway because I'm not sure if my proof is correct.
     
    Last edited: Nov 24, 2013
  2. jcsd
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