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Math Help - Proof of Pratt's Lemma

  1. #1
    Junior Member
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    Proof of Pratt's Lemma - Lebesgue Integration

    Hello!

    Could you please help me with this question:

    Prove the following (known as Pratt's Lemma):

    Let  \{f_k\}_{k\geq 1} , \{g_k\}_{k\geq 1} , \{G_k\}_{k\geq 1} be sequences of integrable functions. If

    1)  f_k (x) \rightarrow f(x), g_k (x) \rightarrow g(x), G_k (x) \rightarrow G(x), as  k \rightarrow \infty \forall x
    2)  g_k (x) \leq f_k (x) \leq G_k (x)  \forall k \in \mathbb{N}  \forall x
    3)  \int g_k (x) dx \rightarrow \int g(x) dx and  \int G_k (x) dx \rightarrow \int G(x) dk as  k \rightarrow \infty with  \int g(x) dx and  \int G(x) dx both finite,

    then

     lim_{k \rightarrow \infty} \int f_k (x) dx = \int f(x) dx and  \int f(x) dx is finite.

    I was thinking about mimicking the proof of the Dominated Convergence Theorem (noting that  f_k (x) - g_k (x) \geq 0 and by Fatou's Lemma, concluding that  \int (f_k-g)(x) \leq \liminf_{n \rightarrow \infty} \int (f_k - g_n) (x) dx ) but I'm not quite sure where to go from there... Any hints/help is appreciated
    Last edited by Mimi89; March 2nd 2010 at 01:37 PM. Reason: Correcting misprints, again
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  2. #2
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    Joined
    May 2010
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    Iam not sure of the proof.

    You can directly use DCT to prove this. given that function is bounded and has a pointwise convergence, first prove that f belongs to L' and lfl belongs to L'. that proof is obvious,as the functions are positive Then proceed with your method,use Fatou's lemma here both sup and inf and given the condition of function g, take limits k to infinity on both sides, and use the condition convergence of E(g), and you get the convergence of E(f).

    I hope it helps
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  3. #3
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    apply every convergent sequence is a cauchy sequence and every cauchy sequence is bounded and then DCT is directly applicable.
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