11.3 Riemann-Stieltjes Sums and Integrals

Definition 11.3.1 (Riemann-Stieltjes Sum, [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, and $G: [a, b] \to E_{2}$.

Let $f: [a, b] \to E_{1}$ and $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_{t}([a, b])$, then

is the Riemann-Stieltjes sum of $f$ with respect to $G$ and $(P, c)$.

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

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 $G$.

Lemma 11.3.3 (Summation by Parts, [Proposition 1.4, 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, $f: [a, b] \to E_{1}$, $G: [a, b] \to E_{2}$, and $(P, c) \in \scp_{t}([a, b])$, then

where $P' = \seqfz[n+1]{y_j}= [a, c_{1}, \cdots, c_{n}, b]$ and $c' = \seqf[n+1]{d_j}= [x_{0}, \cdots, x_{n}]$.

Theorem 11.3.4 (Integration by Parts). Let $[a, b] \subset \real$, $E_{1}, E_{2}, H$ be TVSs over $F \in \RC$, and $E_{1} \times E_{2} \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map.

Let $f: [a, b] \to E_{1}$ and $G: [a, b] \to E_{2}$, then $f \in RS([a, b], G)$ if and only if $G \in RS([a, b], f)$. In which case,