Definition 5.3.2 (Riemann-Stieltjes Integral).label Let $[a,b]\suf\R$, $E,F,H$ be TVSs over $K\in \curl{\R,\com}$, $G:[a,b]\to F$ and suppose there exists a continuous bilinear map $E\ti F\ni (x,y)\mto xy\in H$. Let $f:[a,b]\to E$ then $f$ is Riemann-Stieltjes integrable with respect to $G$ if the limit
\begin{align*}\integral{a}{b}fdG=\integral{a}{b}f(t)G(dt)=\limit{(P,c)\in\scr{P}_t([a,b])}S(P,c,f,G)\end{align*}
exists. In which case we call the Riemann-Stieltjes integral of $G$. The set $RS([a,b],G)$ of all Riemann-Stieltjes integrable functions with respect to $G$ is a vector space.