I'm trying to prove the following claim:
If $f \in L^1([a,b])$ and $\forall x\in [a,b], \int_a^x f(u)du = 0$ then $f=0$ almost everywhere (a.e.).
I thought of using the definition of $\int f = \int f^+ - \int f^-$, with the goal of applying the following known fact for functions in $L^+$ (both $f^+$ and $f^-$ are in $L^+$):
If $f \in L^+$ and $\int f = 0$ then $f=0$ a.e.
But I cannot immediately use this since I only have that their difference is 0 (i.e. $\int f^+ - \int f^- = 0$), and not that each one of them is 0.
Any ideas on how I could proceed? (ideally using basic results in Real Analysis - e.g. convergence theorems like DCT, results on $L^+$, basic results on $L^1$ etc.).