Let $f:[0,1]\rightarrow \mathbb{R}$ be a bounded measurable function. We say $t\in[0,1]$ is a leader if $\int_0^{\epsilon} f(t+s)\,\mathrm{d}s<0$ for some $\epsilon\in [0,1-t]$. Let $L\subset [0,1]$ be the set of leaders. Then is it true that $\int_L f(t)\,\mathrm{d}t\leq 0$?
A discrete analogue of this problem is true. See for example, http://www.unige.ch/math/folks/karlsson/kama.pdf page 114