Show that $L^1$ function is $0$ a. e.

Question

Suppose $f \in L_1(\mathcal{R})$ satisfies for every measurable $A \subset \mathcal{R}$

$$ |\int_A f| \leq [m(A)]^{(1+\epsilon)} $$

for some $\epsilon >0$. Prove $f=0$ a.e.

This is a problem on my analysis qual review sheet. I've been trying a proof by contradiction with no avail... Help?

Answer 1

Here is a hint: by the Lebesgue differentiation theorem, the limit $$\lim_{r\to 0} \frac1{m(B_r(x))} \int_{B_r(x)} f(y) dy$$ where $B_r(x)=\{y\,:\,|x-y|<r\}$ exists and equals $f(x)$ for almost every $x$. Can you take it from there?