Lebesgue-Stieljes Measures

Given $F:R\rightarrow R$ nondecreasing and right contiuous, define an outer measure by

$\mu _{F}^{*}(A)=\inf \left\{\sum _{i}\mu _{F}^{*}(\left(a,b\right])\ :\ A\subset \bigcup _{i}\left(a,b\right]\right\}$

where $\mu _{F}^{*}(\left(a,b\right])=F(b)-F(a)$ and the infimum taken over all coverings of A by countably many semiopen intervals. By Carathéodory's Theorem, we know that $\mu _{F}:=\left.\mu _{F}^{*}\right|_{M_{\mu _{F}^{*}}}$ is a measure. This measure is sometimes called the Lebesgue–Stieltjes measure associated with F.