This is a problem from Probability and Stochastics by Erhan Cinlar.
Let $\mu$ be a measure on $(\mathbb{R}_{+}, \mathcal{B}(\mathbb{R}_{+}))$ such that $c(t)=\mu[0,t]$ is finite for every $t$ in $\mathbb{R}_{+}$. If $\mu$ is absolutely continuous with respect to the Lebesgue measure $\lambda$ on $\mathbb{R}_{+}$, (prove) then the cumulative distribution function $c$ is differentiable at $\lambda$-almost every $t$ in $\mathbb{R}_{+}$ and
$p(t)=\frac{\mu(dt)}{\lambda(dt)}=\frac{d}{dt}c(t)$ for $\lambda$-almost every $t$.
The conclusion seems pretty natural to me, but I am stuck with how to prove that $c$ is differentiable at $\lambda$-almost every $t$ in $\mathbb{R}_{+}$; we have $\lambda(p \mathbf{1}_{[t, t+\epsilon_i]})=c(t+\epsilon_i)-c(t)$ for strictly decreasing sequence $\{\epsilon_i\}$ (which converges to $0$), and thus by definition we have $\min_{[t, t+\epsilon_i]}p \leq \frac{c(t+\epsilon_i)-c(t)}{\epsilon_i} \leq \max_{[t, t+\epsilon_i]}p$. So with this (somewhat naive) approach for me it seems that for $c(t)$ to be differentiable we should have that $p$ is discontinuous only at negligible sets; but I'm not sure if this is the right way. Since I am a beginner at measure theory, any help would be greatly appreciated.