Results 1 to 3 of 3

Thread: Proof of Pratt's Lemma

  1. #1
    Junior Member
    Dec 2009

    Proof of Pratt's Lemma - Lebesgue Integration


    Could you please help me with this question:

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

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

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


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

    I was thinking about mimicking the proof of the Dominated Convergence Theorem (noting that $\displaystyle f_k (x) - g_k (x) \geq 0 $ and by Fatou's Lemma, concluding that $\displaystyle \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; Mar 2nd 2010 at 01:37 PM. Reason: Correcting misprints, again
    Follow Math Help Forum on Facebook and Google+

  2. #2
    May 2010
    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
    Follow Math Help Forum on Facebook and Google+

  3. #3
    May 2010
    apply every convergent sequence is a cauchy sequence and every cauchy sequence is bounded and then DCT is directly applicable.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Euclid's Lemma Proof by Induction
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: Feb 7th 2011, 01:25 AM
  2. Lemma proof help?
    Posted in the Geometry Forum
    Replies: 0
    Last Post: Mar 20th 2010, 03:08 PM
  3. Pumping Lemma proof...
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: Mar 15th 2010, 07:50 AM
  4. Proof of Lemma on Polynomials
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Sep 30th 2009, 11:02 AM
  5. Proof of a lemma
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: Apr 29th 2009, 05:19 AM

Search tags for this page

Click on a term to search for related topics.

Search Tags

/mathhelpforum @mathhelpforum