A nice property of $L_1$ function in $[0,1]$. Let $f$ is in $L^1([0,1],m)$, $m$ is a Lebesgue measure and suppose that $f(x)>0$ for all $x$. Show that for any $ 0<\epsilon<1 $ there exists $\delta$ >0 so that $\int_{E} f(x)dx\ge \delta$ for any set $E\subset[0,1]$ with $m(E)=\epsilon$
I have done a similar problem something like  if $f$ is in $L^1$ then we have $\epsilon$ and $\delta$ definition. However, this problem seems a bit different even kinda of opposite in the sense of $\epsilon$ and $\delta$. I could get my head around it.

Note that all the conditions in Lebesgue integral of non-negative
are satisfied.
You may refer to my solution.
