Let $f$ be a Lebesgue integrable function on $[0,2]$. If $\int_E fdx=0$ for all measurable set $E$, such that $m(E)=\pi/2$. Is $f=0$ a.e. Prove or disprove
I could not figure out anything. Can a function be very oscillatory so that on every interval its integral is zero?
Any hint and approach are welcome.
Hint: let $A$ and $B$ be small enough (let’s say $m(A)=m(B)<0.001$). Then there exists $C$ disjoint from $A \cup B$ such that $m(C)=\pi/2 – m(A)$. Using your conditions on $A \cup C$ and $B \cup C$ show that $\int_Afdm=\int_Bfdm$. Now consider the positivity and negativity sets of $f$. Show, that if one or both of them have measure zero you are done. Show that otherwise it will contradict previous argument