I'm reading through a paper and I'm having trouble following the logic of the following step.
"Suppose $\psi:(0,\infty) \to [0,\infty)$ is a non-negative, non-decreasing function. Let $\Psi$ be the primitive function of $\psi$, i.e. $\Psi' = \psi$."
Can we actually do this? I believe monotonicity of $\psi$ means that an antiderivative can indeed be defined, and would be continuous, but wouldn't it only be differentiable almost everywhere? In particular, at points of discontinuity of $\psi$ (the set of which I believe will have measure zero, again by monotonicity) we cannot say $\Psi' = \psi$?
In summary, is it instead true that $\Psi' = \psi$ a.e.?
Thank you for your time!