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:
∫R∖E|f|≥ϵ
Since ϵ>0 was arbitrary then we have:
∫R∖E|f|=∞
But μ(R∖E)≥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)|
so there exists some n0 such that
∫χEn0|f|>∫|f|−ϵ.
Note that
∫χEn0|f|=∫En0|f|,
and |En0|≤2n0<∞.
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