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Properties of Lebesgue Integral

  1. Oct 13, 2014 #1
    1. The problem statement, all variables and given/known data
    Let ##f,g## be non-negative Lebesgue measurable functions, and let ##E## be a measurable set. Prove that:

    ##f\ge g\implies \int_E f d\lambda \le \int_E g d\lambda##

    2. Relevant equations


    3. The attempt at a solution
    I've been trying to apply Fatou's lemma but haven't been getting anywhere.
     
  2. jcsd
  3. Oct 13, 2014 #2
    Here's a little more detail on my attempt
    ## \int (f-g)\le liminf_{n\to\infty}\int{(f_n-g_n)}##
    ##\implies \int{f}-\int{g}\le liminf_{n\to\infty}\int{f_n}-liminf_{n\to\infty}\int{g_n}##
    ##\implies -\int g -limsup_{n\to \infty}\int{-g_n} \le -\int f -limsup_{n\to \infty}\int{-f_n} ##
    ##\implies \int g +limsup_{n\to \infty}\int{g_n} \ge \int f +limsup_{n\to \infty}\int{f_n} ##
    ##\implies \int g \ge \int f##

    My confusion arises when I compare this to monotonicity. I must be missing a step and it must be because I'm not using the fact that
    ##\int_E f=\int{f \mathcal{X}_E}##
     
  4. Oct 13, 2014 #3

    Ray Vickson

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    You are getting nowhere because the result is false. Try the example ##f(x) = 2, g(x) = 1 \; \forall x## and ##E = [1,2]##.
     
  5. Oct 13, 2014 #4

    Fredrik

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    I think one of those inequalities is supposed to point in the other direction. For example, if ##f\leq g##, then ##\int_E f\mathrm d\lambda\leq \int_E g\mathrm d\lambda##. There's more than one definition of the Lebesgue integral, so you will have to tell us which one you're using. Is ##\lambda## the Lebesgue measure on ##\mathbb R^n##, or just some arbitrary measure?

    Regardless of what definition you're using, it will probably be slightly easier to break this up into two parts:
    (a) If ##f\geq 0## then ##\int_E f\mathrm d\lambda\geq 0##.
    (b) If ##f\leq g## then ##\int_E f\mathrm d\lambda\leq \int_E g\mathrm d\lambda##.

    (a) is slightly easier to prove than (b) alone, and if you have proved (a), it's very easy to use it to prove (b).
     
  6. Oct 13, 2014 #5
    Ack! That's not good, this was the professors practice problem for the upcoming test. Also, I double checked and the problem is stated here as he gave it to the class. Please disregard my request for assistance.
     
    Last edited: Oct 13, 2014
  7. Oct 13, 2014 #6

    Fredrik

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    Then there's a typo in the problem. If you reverse the direction of one of the inequalities, you get one of the standard theorems that's proved in every book on integration.
     
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