A locally finite measure $\mu$ is a measure such that for all $x \in \mathbb{R}$, there exists $\epsilon > 0$ such that $\mu(]x-\epsilon;x+\epsilon]) < + \infty$.
My goal is to show that a measure $\mu$ of $(\mathbb{R}, B(\mathbb{R}))$ which is locally finite is a Stieltjes measure.
I started by showing that for any compact K of $\mathbb{R}, \mu(K) < + \infty$, thanks to the property of Borel-Lebesgue. Does this help in any way?
I have no idea how to proceed.