I am having trouble reading the proof of the construction of the Lebesgue-Stieltjes measure in Terence Tao's An Introduction to Measure Theory. See Theorem 1.7.9 on pp. 189-192 on https://terrytao.files.wordpress.com/2011/01/measure-book1.pdf.
Here are my questions:
What is the functional form of the $F-$volume of an interval? If $I$ is, say, $(a,b)$, $- \infty < a < b < \infty$, then is it $F(b) - F(a)$? If so, is it the case for all intervals, closed, open, half open-half closed, half infinite etc.?
If 1 is true, then the formulas for the Borel measures of the intervals given in the stem of Theorem 1.7.9 don't make much sense.
I think I can pretty much follow the remainder of the proof, but I am very unsure of how an algebra is set up, and how a premeasure is defined on the said algebra, which is then used to exploit the Hahn-Kolmogorov Theorem.
The $F$-volume of $I$ is defined by the equalities in $(1.33)$. If I have to guess, what might be confusing you is that you imagine that $F$ is continuous. But $F$ may have jumps, and where $F$ has a jump, the measure will have an atom.
For instance, take $F=1_{[1,\infty)}$, so there's a jump at $1$. Now $$ |[0,1]|_F=F_+(1)-F_-(0)=1-0-1, $$ while $$ |[0,1)|_F=F_-(1)-F_-(0)=0-0=0. $$ The measure $\mu_F$ will have an atom at $1$, i.e. $\mu_F(\{1\})=1$.