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 the measure Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: Double subscripts: use braces to clarify"): {\displaystyle \mu _{F}:=\mu _{F}^{*}\mid _{M}_{\mu }_{F}^{*}} arising from the outer measure $\mu _{F}:=mu_{F}^{*}$ \left.\frac{\partial f}{\partial x}\right|_{\hat x_{k-1}}