Given that $F$ is a left continuous and also montonic function $\mathbb R \rightarrow \mathbb R$ and $\Lambda $ is the Lebesgue-Stieltjes measure which is induced by $F$, how can we calculate the measures of the following sets:
$\Lambda(\{x\}),\Lambda([a,b])$ and $\Lambda((a,b)) $?
So, $$\Lambda(A) = \inf\left\{\sum_{k=1}^{\infty} \lambda([a_k,b_k));\,A \subset \bigcup \limits_{k=1}^{\infty}[a_k,b_k)\right\}$$
where $\lambda([a,b))= F(b) - F(a)$ if $b \geq a$ and $0$ else.
So if I understand it correctly, if I take a point as this set $A$, I just have to find the infimum of a cover (here the union of semi-open intervals) for this point and then just calculate $F(b) - F(a)$ So for $\Lambda(\{x\})$ couldn't I just use as cover such a thing as $$\bigcup \limits_{k=1}^{\infty} [x,2^{-k}\cdot\varepsilon]?$$