Let $f:[a,b]\to \mathbb R$ be a measurable function .Then
Prove that if $\int _c ^d f(x)\operatorname {dx}=0$ for all $a\le c <d\le b$ then $f=0 \operatorname{a.e.}$
My try: Let $A=\{x:f(x)\neq 0\}$ .Define $A^+=\{x:f(x)>0\} $ and $A^{-}=\{x:f(x)<0\}$
We consider only the set $A^+=\{x:f(x)>0\} $ for $A^-$ we consider $-f$. Surely $A^{+}$ is a measurable set.,
How should I proceed from here?
Please give some hints