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Thread: Lebesgue Integral

  1. #1
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    Lebesgue Integral

    Suppose $\displaystyle f$ is a non-negative integrable function on $\displaystyle X$.For each $\displaystyle n$,define $\displaystyle E_n= \{ x \in X : f(x)>n \}$.
    Show that $\displaystyle lim_{n \rightarrow \infty} m(E_n) =0$.
    ($\displaystyle m(A)$ is the measure of $\displaystyle A$.)

    I was trying to prove that since $\displaystyle f$ is integrable,then $\displaystyle f < \infty a.e.$.Since $\displaystyle E_n$ is a decreasing sequence of sets,so $\displaystyle lim_{n \rightarrow \infty} m(E_n) = m( \bigcap_{n \in N} E_n)$
    Since $\displaystyle f< \infty a.e$ so $\displaystyle \bigcap_{n \in N} E_n = \phi$

    so $\displaystyle m( \phi ) = 0 $ and hence get the result.
    but the problem is I have to assume $\displaystyle m(E_1) < \infty$ to claim the result.I was stuck here.
    Can anyone help?Or if this is not the correct approach,can anyone show me the right way?
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  2. #2
    Moo
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    Hello,

    What do you think of that ?

    I first started by looking at the integral of f over $\displaystyle E_n$ :

    $\displaystyle \int_{E_n} f ~d\mu=\int_{\mathbb{R}} f \cdot \bold{1}_{\{f>n\}} ~d\mu$

    But in this case, we have $\displaystyle f>n$. Otherwise, it's 0.
    So $\displaystyle \int_{E_n} f~d\mu>\int_{\mathbb{R}} n \cdot \bold{1}_{\{f>n\}} ~ d\mu=n\int_{\mathbb{R}} \bold{1}_{\{f>n\}} ~d\mu=n\mu(E_n)$

    Now, we know that f is integrable. This means that $\displaystyle \int_{\mathbb{R}}f ~d\mu<\infty$

    And since $\displaystyle E_n\subset \mathbb{R}$, we have $\displaystyle \int_{\mathbb{R}}f ~d\mu\geq \int_{E_n}f ~d\mu$


    Finally, we get $\displaystyle \infty > \int_{\mathbb{R}} f ~d\mu> n\mu(E_n)$

    Now what happens if n goes to infinity ? Unless $\displaystyle \mu(E_n)\to 0$, $\displaystyle n\mu(E_n) \to \infty$...
    A simple proof by contradiction can finish it
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