I would like to show that it's possible to associate to a measure a monotone increasing right-continuous function s.t.: $\mu(\left(a,b\right])=F(b)-F(a)$.
How can I prove that a function like
$$F(x)=\mu(\left(0,x\right]) ~~ x>0$$
$$F(x)=0 ~~ x=0$$
$$F(x)=-\mu(\left(x,0\right])~~ x<0$$
is right-continuous? I thought about considering a sequence $x_n$ decreasing to $x$ such that $\lim(\left(x,x_n\right])=\emptyset$, $\lim \>\mu(\left(x,x_n\right])=0$ and then directly express all in terms of $F$ . My doubts concern the fact that there is an hint for the exercise saying: consider the countable additivity property of the measure in order to show that the function is right continuous.