TITLE: What is so special about the Lebesgue-Stieltjes measure
QUESTION [7 upvotes]: A measure $\lambda: B(\mathbb{R}^n) \rightarrow  \overline{{\mathbb{R_{\ge 0}}}}$ that is associated with a monotone increasing and right-side continuous function $F$ is called a Lebesgue-Stieltjes measure. But I am wondering, why it is not true that every measure $\lambda: B(\mathbb{R}^n) \rightarrow  \overline{{\mathbb{R_{\ge 0}}}}$ is a Lebesgue-Stieltjes measure?

REPLY [2 votes]: If you have a measure $\mu$ in $\mathbb{R}$ that is finite in compact sets then we define the function 
$$
  F(x) = \begin{cases}
     \mu((0,x])&\text{if $x > 0$} \\
    -\mu((x,0]) &\text{if $x < 0 $} \\
    0 &\text{if $x=0$}
  \end{cases}
$$
Then $F$ is increasing càdlag function and $\mu = \mu_{F}$.
This also shows that every measure in $\mathbb{R}$ which are finite in compact sets is regular.