Definition of Lebesgue-Stieltjes measure on $\mathbb R$

Question

Let $F:\mathbb R\to\mathbb R$ be a non-decreasing, left-continuous function. Let $a,b\in\mathbb R$, then define the Lebesgue-Stieltjes measure $$ m[a,b]=F(b+)-F(a), \quad m(a,b)=F(b)-F(a+) $$ $$ m(a,b]=F(b)-F(a+), \quad m[a,b)=F(b)-F(a) $$

I am wondering what the notation F(a+) or F(b+) means.

Since $F$ is left-continuous we could have, e.g. $F(x)=\begin{cases} 1 & ,x\le3\\ 2 & ,x>3\end{cases}$.
Now is $F(3+)=1$ or $F(3+)=2$ ?

Intuitively I would assume that e.g. $$m[3,4]=m(3,4)=m(3,4]=m[3,4)=F(4)-F(3)=2-1=1$$ in any case. But I am probably wrong.

Answer 1

The notation $f(a+)$ means "right hand limit," i.e., $\lim_{x\downarrow a} f(x)$. Similarly you have $$ f(a-) = \lim_{x\uparrow a} f(x).$$ These limits always exist for a cumulative distribution function since such function have a range contained in $[0,1]$ and are monotone nondecreasing.

In your example, you have $F(3-) = 1$ and $F(3+) = 2.$ Since $F(3) = 1$, $F$ is left-continuous at 3. Remember, the limits disregard behavior at the target point.

Answer 2

By definition \begin{align} F(a^+)\triangleq \lim_{\substack{x\to a \\ a<x }}F(x) \end{align} that is, the number $F(a^+)$ is the number such that $$ \Big(\forall \;\epsilon>0 \Big)\Big(\exists\; \delta>0 \Big) \bigg[\Big(x\in D_F \Big)\wedge \Big(0<x-a<\delta\Big) \implies \big|F(x)-F(a^+) \big|<\epsilon \bigg] $$