Let $f ∈ L_1([0,1])$ be a function such that $\int_E f(x)dx = 0$ for any measurable set $E ⊂ [0,1]$ of Lebesgue measure $0.99.$
If $f$ is Lebesgue integrable on $[0,2]$ and $\int_E fdx=0$ for all measurable set E such that $m(E)=\pi/2$. Prove or disprove that $f=0$ a.e.