# Let's say we're given a Lebesgue-Stieltjes measure &#x03BC;<!-- μ --> and an associated functi

Let's say we're given a Lebesgue-Stieltjes measure $\mu$ and an associated function ${F}_{\mu }$ such that
$\mu \left(\left[a,b\right)\right)={F}_{\mu }\left(b\right)-{F}_{\mu }\left(a\right)$
for all semi-intervals in $\mathbb{R}.$
I aim to calculate $\mu \left(\left[a,b\right]\right),\mu \left(\left(a,b\right)\right),\mu \left(\left(a,b\right]\right)$ in terms of the function ${F}_{\mu }.$
My current idea is to express the closed, open, and semi-interval in terms of the left semi-interval as such:
$\left(a,b\right)=\bigcup _{i=1}^{\mathrm{\infty }}\left[{a}_{i},b\right)$
where $a<{a}_{i+1}<{a}_{i}$ and $\underset{n\to \mathrm{\infty }}{lim}{a}_{n}=a$. For example, such a sequence could be ${a}_{i}=a+{2}^{-i}.$
Then,$\mu \left(\left(a,b\right)\right)=\sum _{i=1}^{\mathrm{\infty }}\mu \left(\left[{a}_{i},b\right)\right)=\sum _{i=1}^{\mathrm{\infty }}\left({F}_{\mu }\left(b\right)-{F}_{\mu }\left(a+{2}^{-i}\right)\right).$
We could get similar expressions using $\left(a,b\right]=\left[a-ϵ,b\right]╲\left[a-ϵ,a\right]$ and then substituting as needed.
However, all of these expressions would be extremely messy and would give very complicated answers. Is there a cleaner way of expressing the measure of each interval?
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Nathen Austin
$\mu \left(\left(a,b\right)\right)=\underset{n\to \mathrm{\infty }}{lim}\mu \left(\left[a+\frac{1}{n},b\right)\right)=\underset{n\to \mathrm{\infty }}{lim}\left[{F}_{\mu }\left(b\right)-{F}_{\mu }\left(a+\frac{1}{n}\right)\right]$. This is ${F}_{\mu }\left(b\right)-{F}_{\mu }\left(a+\right)$ where ${F}_{\mu }\left(x+\right)$ denotes the right-hand limit of ${F}_{\mu }\left(b\right)$ at $x$.
Similarly, ${F}_{\mu }\left(\left[a,b\right]\right)={F}_{\mu }\left(b+\right)-{F}_{\mu }\left(a\right)$ and ${F}_{\mu }\left(\left(a,b\right]\right)={F}_{\mu }\left(b+\right)-{F}_{\mu }\left(a+\right)$.
[It should be noted that ${F}_{\mu }$ is continuous from the left at every point and, as a monotone function, has right hand limit at every point].