Definition 11.3.2 (Riemann-Stieltjes Integral, [Section X.1, Lan93]). Let $[a, b] \subset \real$, $E_{1}, E_{2}, H$ be TVSs over $F \in \RC$, $E_{1} \times E_{2} \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, $G: [a, b] \to E_{2}$.
Let $f: [a, b] \to E_{2}$, then $f$ is Riemann-Stieltjes integrable with respect to $G$ if the limit
\[\int_{a}^{b} f dG = \int_{a}^{b} f(t)G(dt) = \lim_{(P, c) \in \scp_t([a, b])}S(P, c, f, G)\]
exists. In which case, $\int_{a}^{b} fdG$ is the Riemann-Stieltjes integral of $G$.
The set $RS([a, b], G)$ is the vector space of all Riemann-Stieltjes integrable functions with respect to $G$.