If $\mu$ is a measure on $(\mathbb{R}_+, \mathcal{B}(\mathbb{R}_+))$ such that $\mu([0,t])$ is finite for all $t\in\mathbb{R}_+$, where $\mathbb{R}_+=[0,+\infty)$ and $\mathcal{B}(\mathbb{R}_+)$ is Borel $\sigma$-algebra on $\mathbb{R}_+$. $\lambda$ is Lebesgue measure on $\mathbb{R}_+$. $\mu\ll\lambda$.
I need to show the following limit at $\lambda$-almost every $t$ in $\mathbb{R}_+$:
$\lim\limits_{\epsilon\to0^+}\frac{1}{\epsilon}\int_{(t,t+\epsilon]}pd\lambda=p(t)$,
where $p$ is the Radon-Nikodym derivative $d\mu/d\lambda$.
The equation is familiar to me in calculus, but here, as a Radon-Nikodym derivative, all I know about $p$ is that it is positive measurable, not a bounded real-valued one in the definition of Riemann integral, so the tools I have in calculus cannot be applied (a similar thread is here, but it's all calculus). From the condition that $\mu([0,t])$ is finite, I have a feeling that $p$ is smooth enough to make the limit hold, but I don't know specifically what attribute of a Radon-Nikodym derivative can be used to prove this limit. Can you please give me some help? Thanks a lot.