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Math Help - Two Lebesgue Integration Problems

  1. #1
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    Two Lebesgue Integration Problems

    1. Let f be a nonnegative measurable function on \Re. Show that if F : \Re \rightarrow \Re is defined by \int_{(0,x]} f(t)dm(t) then F is continuous on \Re.

    I know that to prove that a function is continuous, you have to show that for any open set A \subseteq \Re so that f^{-1}(A) is open, but I'm not sure what to do from there.

    2. Let f be a nonnegative measurable function on a measurable set E with \int_E f(x)dm(x)<\infty. Prove that for each \epsilon>0 there exists \delta>0 such that for every measurable set A \subseteq E with m(A) < \delta we have \int_A f(x)dm(x) < \epsilon

    I am clueless how to even start this problem.
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  2. #2
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    For 1. use the sequential characterization of continuity. Take a sequence {x_n} converging to x. It suffices to show F(x_n) --> F(x).
    A hint to do this is bring (0,x_n] in as a characteristic function and use an appropriate convergence theorem.
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  3. #3
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    Quote Originally Posted by grad444 View Post
    2. Let f be a nonnegative measurable function on a measurable set E with \int_E f(x)dm(x)<\infty. Prove that for each \epsilon>0 there exists \delta>0 such that for every measurable set A \subseteq E with m(A) < \delta we have \int_A f(x)dm(x) < \epsilon

    I am clueless how to even start this problem.
    For 2.: Let \varepsilon>0. Using the bounded convergence theorem, prove that there exists M such that \int_{\{f>M\}} f(x)dm(x)<\varepsilon/2 (integration on the set where f(x)>M). Then choose \delta<\varepsilon/(2M), and conclude using:

    \int_A f(x)dm(x)= \int_{A\cap\{f>M\}} f(x)dm(x)+\int_{A\cap\{f\leq M\}} f(x)dm(x)\leq \varepsilon/2 + M m(A).
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