Lebesgue measure vs Lebesgue-Stieltjes measure Reading advanced probability theory book I've come across Lebesgue-Stieltjes measure. Could someone explain what is the difference between it and "standard" Lebesgue measure on $\mathbb{R}$? Thank you.

It can be proven that for any increasing right-continuous function $F:\mathbb{R}\rightarrow \mathbb{R}$ there exists a unique measure $\mu_F$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ satisfying the property:
$$ \mu_F((a,b])=F(b)-F(a) $$
for every interval (a,b] with $a<b$.
$\mu_F$ is called the Lebesgue-Stieltjes measure belonging to F. The Lebesgue measure is simply the Lebesgue-Stieltjes measure belonging to the identity function.
