Lebesgue Integral

• Jan 29th 2010, 06:37 AM
problem
Lebesgue Integral
Let $\displaystyle f \in L_p(X)$,1 $\displaystyle \le p < \infty$ and let $\displaystyle \epsilon > 0$. Show that there exists a set $\displaystyle E_\epsilon \subseteq X$ with $\displaystyle m(E_\epsilon) < \infty$ such that if $\displaystyle F \subseteq X$ and $\displaystyle F \cap E_\epsilon = \phi$,then $\displaystyle \int_F |f|^p dm < \epsilon^p$

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

$\displaystyle \int_F |f|^P dm < \int_F (\delta_\epsilon)^p dm = (\delta_\epsilon)^p m(F)$
I am not sure whether my construction of the sets are correct because I can not make any conclusion regarding $\displaystyle m(F)$.
Can anyone comment on this?
• Jan 29th 2010, 01:05 PM
Opalg
Quote:

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

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

$\displaystyle \int_F |f|^P dm < \int_F (\delta_\epsilon)^p dm = (\delta_\epsilon)^p m(F)$
I am not sure whether my construction of the sets are correct because I can not make any conclusion regarding $\displaystyle m(F)$.
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 $\displaystyle |f|^p$, 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 $\displaystyle |f|^p$. If the integral $\displaystyle \int_X|f|^pdm$ is finite (as it is if $\displaystyle f \in L_p(X)$) then each of these simple functions must have finite integral and therefore finite support. We can find a simple function s such that $\displaystyle s\leqslant|f|^p$ and $\displaystyle \int_Xs\,dm>\int_X|f|^pdm - \varepsilon^p$. Now take $\displaystyle E_\varepsilon$ to be the support of s. You should find that this does the required job.