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Math Help - Lebesgue Integral

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    Lebesgue Integral

    Let f \in L_p(X),1 \le p < \infty and let \epsilon > 0. Show that there exists a set E_\epsilon \subseteq X with m(E_\epsilon) < \infty such that if F \subseteq X and F \cap E_\epsilon = \phi,then \int_F |f|^p dm < \epsilon^p


    I let E_\epsilon = \{ x \in X : |f(x)| \ge \delta_\epsilon \},where \delta_\epsilon > 0 and F = \{ x \in X : |f(x)| < \delta_\epsilon \}.

    \int_F |f|^P dm < \int_F (\delta_\epsilon)^p dm = (\delta_\epsilon)^p m(F)
    I am not sure whether my construction of the sets are correct because I can not make any conclusion regarding m(F).
    Can anyone comment on this?
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  2. #2
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    Quote Originally Posted by problem View Post
    Let f \in L_p(X),1 \le p < \infty and let \epsilon > 0. Show that there exists a set E_\epsilon \subseteq X with m(E_\epsilon) < \infty such that if F \subseteq X and F \cap E_\epsilon = \phi,then \int_F |f|^p dm < \epsilon^p


    I let E_\epsilon = \{ x \in X : |f(x)| \ge \delta_\epsilon \},where \delta_\epsilon > 0 and F = \{ x \in X : |f(x)| < \delta_\epsilon \}.

    \int_F |f|^P dm < \int_F (\delta_\epsilon)^p dm = (\delta_\epsilon)^p m(F)
    I am not sure whether my construction of the sets are correct because I can not make any conclusion regarding m(F).
    Can anyone comment on this?
    I think you have to go right back to the definition of the Lebesgue integral to do this properly. For a positive function such as |f|^p, the usual way to define its integral (as described here, for example) is that it is the supremum of the integrals of nonnegative simple functions majorised by |f|^p. If the integral \int_X|f|^pdm is finite (as it is if f \in L_p(X)) then each of these simple functions must have finite integral and therefore finite support. We can find a simple function s such that s\leqslant|f|^p and \int_Xs\,dm>\int_X|f|^pdm - \varepsilon^p. Now take E_\varepsilon to be the support of s. You should find that this does the required job.
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