Let's say we're given a Lebesgue-Stieltjes measure $\mu $ and an associated function ${F}_{\mu}$ such that

$\mu ([a,b))={F}_{\mu}(b)-{F}_{\mu}(a)$

for all semi-intervals in $\mathbb{R}.$

I aim to calculate $\mu ([a,b]),\mu ((a,b)),\mu ((a,b])$ 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:

$(a,b)=\bigcup _{i=1}^{\mathrm{\infty}}[{a}_{i},b)$

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,$\mu ((a,b))=\sum _{i=1}^{\mathrm{\infty}}\mu ([{a}_{i},b))=\sum _{i=1}^{\mathrm{\infty}}({F}_{\mu}(b)-{F}_{\mu}(a+{2}^{-i})).$

We could get similar expressions using $(a,b]=[a-\u03f5,b]\u2572[a-\u03f5,a]$ 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?

$\mu ([a,b))={F}_{\mu}(b)-{F}_{\mu}(a)$

for all semi-intervals in $\mathbb{R}.$

I aim to calculate $\mu ([a,b]),\mu ((a,b)),\mu ((a,b])$ 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:

$(a,b)=\bigcup _{i=1}^{\mathrm{\infty}}[{a}_{i},b)$

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,$\mu ((a,b))=\sum _{i=1}^{\mathrm{\infty}}\mu ([{a}_{i},b))=\sum _{i=1}^{\mathrm{\infty}}({F}_{\mu}(b)-{F}_{\mu}(a+{2}^{-i})).$

We could get similar expressions using $(a,b]=[a-\u03f5,b]\u2572[a-\u03f5,a]$ 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?