The integral of a positive a.e function defined on a cube.

Question

Say we take the cube $Q=[0,1]^d \subset \mathbb{R}^d $ and let f be an integrable function on Q and suppose that f>0 a.e.

Fix any $\epsilon >0$ with $\epsilon \leq m(Q)$

I want to prove that inf{$\int_E f: E$ is measurable $ m(E)\geq \epsilon$}>0.

Answer 1

Hint: using the fact that $f>0$ almost everywhere, find a measurable set $A$ and a $\delta > 0$ such that $m(A) \ge 1-\epsilon/2$ and $f \ge \delta$ on $A$. Then note that if $m(E) \ge \epsilon$, we have $m(E \cap A) \ge \epsilon/2$.