Let $\mu$ be a Lebesgue-Stieltjes measure on $\mathcal{B}_\mathbb{R}$, translation invariant for the class of right half-closed intervals of $\mathbb{R}$, i.e. $\mu(a + I) = \mu(I)$, for all $a \in \mathbb{R}$ and $I = (x, y]$.
I've shown that $\mu\left(\left(0,\frac{p}{q}\right]\right)=\frac{p}{q} \cdot \mu\left(\left(0,1\right]\right)$ is valid for any $\frac{p}{q} \in \mathbb{Q}^{+}$.
I want to use lower or upper continuity of the measure to prove that $\mu((0,x])=x \cdot\mu((0,1])$ any $x\in\mathbb{R}$. How do I proceed ?
Choose sequences $(p_n)_{n\in\mathbb N}$ and $(q_n)_{n\in\mathbb N}$ of integers, $q_n\neq 0$, such that the sequence $\left(p_n/q_n\right)_{n\in\mathbb N}$ is non-increasing and $\lim_{n\to +\infty}p_n/q_n=x$. Since for each $n$, $$\tag{*} \mu\left(\left(0,\frac{p_n}{q_n}\right]\right)=\frac{p_n}{q_n}$$ and $$\left(0,x\right)\subset \bigcup_{n\in\mathbb N}\left(0,\frac{p_n}{q_n}\right]\subset\left(0,x\right]$$
and $\mu\left(\{x\}\right)=0$, we get the wanted conclusion by taking the limit as $n\to +\infty$ in (*).