$f = 0$ a.e. in $[0, T]$ if $\int_0^t f(s) ds = 0$ for all $t \in [0, T]$?
Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y) \, dy = 0$ for all $x$. Then $f = 0$ a.e.