13.3 Riemann-Stieltjes Sums and Integrals
Definition 13.3.1 (Riemann-Stieltjes Sum, [Section X.1, Lan93]).label Let $[a, b] \subset \real$, $E, F, H$ be TVSs over $K \in \RC$, $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, and $G: [a, b] \to F$.
Let $f: [a, b] \to E$ 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 13.3.2 (Riemann-Stieltjes Integral, [Section X.1, Lan93]).label Let $[a, b] \subset \real$, $E, F, H$ be TVSs over $K \in \RC$, $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, $G: [a, b] \to F$.
Let $f: [a, b] \to F$, 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 13.3.3 (Summation by Parts).label Let $[a, b] \subset \real$, $E, F, H$ be TVSs over $K \in \RC$, $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, $f: [a, b] \to E$, $G: [a, b] \to F$, and $(P, c) \in \scp_{t}([a, b])$, then
where $P' = \seqfz[n+1]{y_j}= [a, c_{1}, \cdots, c_{n}, b]$ and $c' = \seqf[n+1]{d_j}= [x_{0}, \cdots, x_{n}]$.
Proof [Proposition 1.4, Lan93]. Denote $c_{0} = a$ and $c_{n+1}= b$, then
$\square$
Theorem 13.3.4 (Integration by Parts).label Let $[a, b] \subset \real$, $E, F, H$ be TVSs over $K \in \RC$, and $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map.
Let $f: [a, b] \to E$ and $G: [a, b] \to F$, then $f \in RS([a, b], G)$ if and only if $G \in RS([a, b], f)$. In which case,
Proof. Suppose that $f \in RS([a, b], G)$. Let $U \in \cn_{H}(0)$, then there exits $P_{0} = \seqfz{x_j}\in \scp([a, b])$ such that $S(P, c, f, G) - \int_{a}^{b} fdG \in U$ for all $(P, c) \in \scp_{t}([a, b])$ with $P \ge P_{0}$. Let
then for any $(Q = \seqfz[m]{y_j}, d = \seqf[m]{d_j}) \in \scp_{t}([a, b])$ with $Q \ge Q_{0}$,
by Lemma 13.3.3, where $d$ and $Q'$ contain $\seqfz{x_j}$. Thus $(Q', d') \ge P_{0}$, and $S(Q', d', f, G) - \int_{a}^{b} fdG \in U$.$\square$
Theorem 13.3.5 (Change of Variables).label Let $[a, b] \subset \real$, $E, F, H$ be locally convex spaces over $K \in \RC$, $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, and $G \in C^{1}([a, b]; F)$, then for any bounded $f \in RS([a, b], G; E)$,
Proof. Let $[\cdot]_{H}: H \to [0, \infty)$ be a continuous seminorm and $[\cdot]_{E}: E \to [0, \infty)$ and $[\cdot]_{F}: F \to [0, \infty)$ be continuous seminorms on $E$ and $F$, respectively, such that for any $x \in E$ and $y \in F$, $[xy]_{H} \le [x]_{E}[y]_{F}$.
Since $G \in C^{1}([a, b]; F)$, $DG \in UC([a, b]; F)$ by Proposition 6.4.5. Thus there exists $\delta > 0$ such that $[DG(x) - DG(y)]_{F} < \eps$ for all $x, y \in [a, b]$ with $|x - y| \le \delta$. Let $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_{t}([a, b])$ with $\sigma(P) \le \delta$, then by the Mean Value Theorem, for each $1 \le j \le n$,
so
and
$\square$
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