Generalized second derivative of a concave and piecewise $C^2$ function

Question

It is mentioned in page 20 of this paper that if $f: \mathbb{R}_+ \to \mathbb R$ is a concave and piecewise $C^2$ function, then the generalized second derivative of $f$ is a signed measure $\mu_f$ such that $\mu_f(\mathrm{d}r) \leq f''(r)\mathrm{d}r$. I am wondering if anyone can help me understand the relation $\mu_f(\mathrm{d}r) \leq f''(r)\mathrm{d}r$. As far as know, a concave function $f$ has the left hand derivative $f'_{-}(x)$ for every $x$, and $f'_{-}$ is left-continuous and non-increasing, so we can define a Borel measure on $\mathbb{R}_+$ by the prescription $$\mu_f([x,y)) = f'_{-}(x) - f'_{-}(y),$$ and we can also denote this measure by $f''(\mathrm{d}x)$. May I know how the authors of the aforementioned paper can claim the sentence in bold letters?