Let $J$ be an open interval and let $f:J\to\mathbb{R}$ be locally integrable. Suppose that for all $0\leq\theta\in C^\infty_c(J)$ it holds $$ \int_J f(t)\,\theta'(t)\,dt\geq 0\,, $$ namely the first distributional derivative of $f$ is non-negative.
Can one conclude that $f$ is non-increasing? Or does one need further assumptions?
The idea is the following: the equation tells us that actually $f'$ is not just a distribution but even a positive measure on $J$. How can I go on from this?