Tuesday, 17 February 2015

real analysis - Lebesgue integral of a function that is measurable, finite almost everywhere, and integrable on a set of finite measure




Let f be measurable, finite a.e, and Lebesgue integrable on R . Show that for all ϵ>0 there exists a measurable set E (i.e μ(E)<) such that:



E|f|>R|f|ϵ



My attempt



Suppose that for all ϵ>0 there exists set of finite measure E such that:



E|f|R|f|ϵ




Then:



RE|f|ϵ



Since ϵ>0 was arbitrary then we have:



RE|f|=



But μ(RE)μ(R)μ(E)=




This contradicts the fact that g is finite a.e.



I don’t think my answer is correct because it seems pretty hand-wavy. Any thoughts on a better way to do this?


Answer



You have assumed the result, which is not a valid proof.



Rather, given ϵ>0, since $E_{n}=\{x\in{\bf{R}}:|x|\leq n,|f(x)|χEn|f||f|,



so there exists some n0 such that
χEn0|f|>|f|ϵ.

Note that
χEn0|f|=En0|f|,

and |En0|2n0<.


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find limh0sin(ha)h without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...