Problem on showing a function is zero a.e.

Let $f\in L^1(\mathbb R)$ be such that$$\int_P f(y) dy =0$$for all Lebesgue measureable sets $P\subset \mathbb R$ with $m(P) = \pi$. Prove that $f(x)=0$ for almost all $x\in \mathbb R$.

In attempt to do this problem, I wanted to partition $\mathbb R$ into intervals of length $\pi$ and then integrate over that partition to get that$$\int |f|=0$$but looking back at the hypothesis, we are only given that $\int_P f =0$, which does not necessarily imply that $\int_P |f|=0$, does it?

I am not sure how to get around this and would appreciate any suggestions in a right direction.