Suppose $F:\mathbb{R}\to\mathbb{R}$ is increasing right continuous and we define a pre-measure on the semi-algebra $\mathcal{A}:=\left\{ \left(a,b\right]\ |\ a\leq b\in\mathbb{R}\right\}$ by $\mu\left(a,b\right]=F\left(b\right)-F\left(a\right)$ . I have easily shown this is finitely additive and I want to show it is also continuous from above to deduce $\sigma$-additivity. So assume $\left(a_{n},b_{n}\right]\in\mathcal{A}$ is descending to $\bigcap_{n=1}^{\infty}\left(a_{n},b_{n}\right]$ . I distinguish between several cases:
If $a_{n+1}>a_{n}$ and $b_{n+1}<b_{n}$ for all $n$ then there exists $c\in\mathbb{R}$ s.t $c\in\left(a_{n},b_{n}\right]$ for all $n$ and also $a_{n}\uparrow c$ and $b_{n}\downarrow c$ . In this case we have $\bigcap_{n=1}^{\infty}\left(a_{n},b_{n}\right]=\left\{ c\right\}$ and so $$\lim_{n\to\infty}\mu_{F}\left(\left(a_{n},b_{n}\right]\right)=\lim_{n\to\infty}\left(F\left(b_{n}\right)-F\left(a_{n}\right)\right)=\lim_{n\to\infty}F\left(b_{n}\right)-\lim_{n\to\infty}F\left(a_{n}\right)=F\left(c\right)-F\left(c-\right)$$
If $b_{n+1}=b_{n}$ for all $n\geq N$ and $a_{n+1}>a_{n}$ for all $n$ then $\bigcap_{n=1}^{\infty}\left(a_{n},b_{n}\right]=\left(c,b_{N}\right]$ where $c=\lim_{n\to\infty}a_{n}$ and so $$\lim_{n\to\infty}\mu_{F}\left(\left(a_{n},b_{n}\right]\right)=\lim_{n\to\infty}\left(F\left(b_{n}\right)-F\left(a_{n}\right)\right)=\lim_{n\to\infty}\left(b_{n}\right)-\lim_{n\to\infty}F\left(a_{n}\right)=F\left(b_{N}\right)-F\left(c-\right)$$
If $b_{n}<b_{n+1}$ for all $n$ and $a_{n+1}=a_{n}$ for all $n\geq N$ then $\bigcap_{n=1}^{\infty}\left(a_{n},b_{n}\right]=\left(a_{N},c\right]$ where $c=\lim_{n\to\infty}b_{n}$ and so $$\lim_{n\to\infty}\mu_{F}\left(\left(a_{n},b_{n}\right]\right)=\lim_{n\to\infty}\left(F\left(b_{n}\right)-F\left(a_{n}\right)\right)=\lim_{n\to\infty}F\left(b_{n}\right)-\lim_{n\to\infty}F\left(a_{n}\right)=F\left(c\right)-F\left(a_{N}-\right)$$
Since I only know $F$ is right continuous I'm constantly stuck with those left sided limits which are preventing me from getting the equalities I want. How exactly do I get around this? Am I supposed to somehow use the fact the discontinuities of $F$ are countable (and of Lebesgue measure zero) and just assume all the points $a_{n},b_{n}$ are continuity points of $F$ ?
Edit: I obviously had a mistake in the first case, there of course doesn't have to be such $c$ and in fact we have $\bigcap_{n=1}^{\infty}\left(a_{n},b_{n}\right]=\left[\sup\left\{ a_{n}\right\} ,\inf\left\{ b_{n}\right\} \right]\notin\mathcal{A}$ which means we can disregard the first case. Similarly in the second case we will also reach a similar contradiction and in general the fact the intersection is in the semi-algebra appears to imply the sequence ${a_{n}}$ must be constant from some place in order for the intersection to be in the semi-algebra. This of course resolves the problem.