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\section{Functions of Bounded Variation}
\label{section:bv}
\begin{definition}[Total Variation]
\label{definition:total-variation}
Let $E$ be a locally convex space, $\rho$ be a continuous seminorm on $E$, $f: [a, b] \to E$, and $P \in \scp([a, b])$ be a partition, then
\[
V_{\rho, p}(f) = \sum_{j = 1}^n \rho(f(x_j) - f(x_{j - 1}))
\]
is the \textbf{variation} of $f$ with respect to $\rho$ and $P$. The supremum over all such partitions
\[
[f]_{\var, \rho} = \sup_{P \in \scp([a, b])}V_{\rho, P}(f)
\]
is the \textbf{total variation} of $f$ on $[a, b]$ with respect to $\rho$.
If $E$ is a normed space, then the variation and total variation of $f$ is taken with respect to its norm.
\end{definition}
\begin{definition}[Bounded Variation, {{\cite[Proposition X.1.1]{Lang}}}]
\label{definition:bounded-variation}
Let $E$ be a locally convex space, $\rho$ be a continuous seminorm on $E$, and $f: [a, b] \to E$. If $[f]_{\var, \rho} < \infty$, then $f$ is of \textbf{bounded variation} with respect to $\rho$.
The space $BV([a, b]; E)$ is the set of functions $[a, b] \to E$ of bounded variation with respect to every continuous seminorm on $E$, and
\begin{enumerate}
\item $BV([a, b]; E)$ is a vector space.
\item For each continuous seminorm $\rho$ on $E$, $[\cdot]_{\var, \rho}$ is a seminorm on $BV([a, b]; E)$.
\item Let $\fF$ be a filter on $BV([a, b]; E)$ and $f: [a, b] \to E$. If
\begin{enumerate}
\item $\pi_x(\fF) \to f(x)$ for all $x \in [a, b]$.
\item For every continuous seminorm $\rho$ on $E$, there exists $U \in \fF$ such that $\sup_{g \in U}[g]_{\var, \rho} = M_\rho < \infty$.
\end{enumerate}
then $f \in BV([a, b]; E)$ with $[f]_{\var, \rho} \le M_\rho$.
\item For any $f \in BV([a, b]; E)$ and continuous seminorm $\rho$ on $E$, $\sup_{x \in [a, b]}\rho(f(x)) \le \rho(f(a)) + [f]_{\var, \rho}$.
\end{enumerate}
If $(E, \norm{\cdot}_E)$ is a normed space, then
\begin{enumerate}
\item[(5)] $f$ has at most countably many discontinuities.
\end{enumerate}
\end{definition}
\begin{proof}
(3): Let $\rho$ be a continuous seminorm on $E$ and $P \in \scp([a, b])$, then by assumption (a),
\[
V_{\rho, P}(f) = \sum_{j = 1}^n \rho(f(x_j) - f(x_{j - 1}))
= \lim_{g, \fF}\sum_{j = 1}^n \rho(g(x_j) - g(x_{j - 1}))
= \lim_{g \in \fF}V_{\rho, P}(g)
\]
By assumption (b), $[0, M_\rho]$ is in the filter generated by $V_{\rho, P}(\fF)$. Thus $V_{\rho, P}(f) \le M_\rho$. As this holds for all $P \in \scp([a, b])$, $V_{\rho, P}(f) \le M_\rho$, and $f \in BV([a, b]; E)$.
(5): For each $n \in \nat$, let
\[
D_n = \bracs{x \in [a, b]|\forall \eps > 0, \exists y \in (x - \eps, x + \eps): \norm{f(x) - f(y)}_E \ge 1/n}
\]
then $D = \bigcup_{n \in \nat}D_n$ is the set of discontinuity points of $f$. If $D$ is uncountable, then there exists $N \in \nat$ such that $D_n$ is infinite.
Fix $N \in \nat$. Let $E_1 = D_n \cap (a, b)$ and $I_1 = (a, b)$, then
\begin{enumerate}
\item[(a)] $|E_k| \ge N - k$.
\item[(b)] $E_k \subset I_k^o$.
\end{enumerate}
for $k = 1$.
Let $k \le N$ and suppose inductively that $E_k, I_k$ have been constructed. Let $x_k \in E_k$, then by (b), there exists $\eps > 0$ such that $[x_k - \eps, x_k + \eps] \subset I_k$ and $|E_k \setminus [x_k - \eps, x_k + \eps]| \ge N - k$. Let $y_k \in [x_k - \eps, x_k + \eps]$ such that $\norm{f(x_k) - f(y_k)} \ge 1/n$, $I_{k + 1} = I_k \setminus [x_k - \eps, x_k + \eps]$, and $E_{k+1} = E_k \setminus [x_k - \eps, x_k + \eps]$, then $I_k$ and $E_k$ satisfies (a) and (b).
Therefore there exists pairs $\bracs{(x_k, y_k)|1 \le k \le N}$ such that $\norm{f(x_k) - f(y_k)}_E \ge 1/n$ for all $n$, and the smallest interval containing each $(x_k, y_k)$ are pairwise disjoint. Thus $[f]_{\var} \ge N/n$ for all $N \in \nat$, so $[f]_{\var} = \infty$.
\end{proof}

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\chapter{The Riemann-Stieltjes Integral}
\label{chap:rs-integral}
\input{./src/fa/rs/partition.tex}
\input{./src/fa/rs/bv.tex}
\input{./src/fa/rs/rs.tex}
\input{./src/fa/rs/rs-bv.tex}

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\section{Partitions}
\label{section:partitions}
\begin{definition}[Partition]
\label{definition:partition-interval}
Let $[a, b] \subset \real$, then a \textbf{partition} of $[a, b]$ is a sequence
\[
P = \seqfz{x_j} = [a = x_0 \le \cdots \le x_n = b]
\]
The collection $\scp([a, b])$ is the set of all partitions of $[a, b]$.
\end{definition}
\begin{definition}[Tagged Partition]
\label{definition:tagged-partition}
Let $[a, b] \subset \real$, then a \textbf{tagged partition} of $[a, b]$ is a pair $(P = \seqfz{x_j}, c = \seqf{c_j})$ such that $c_j \in [x_{j - 1}, x_j]$ for each $1 \le j \le n$.
The collection $\scp_t([a, b])$ is the set of all tagged partitions of $[a, b]$.
\end{definition}
\begin{definition}[Mesh]
\label{definition:mesh}
Let $P$ be a partition of $[a, b] \subset \real$, then
\[
\sigma(P) = \max_{1 \le j \le n}(x_j - x_{j - 1})
\]
is the \textbf{mesh} of $P$.
\end{definition}
\begin{definition}[Fine]
\label{definition:partition-refinement}
Let $P = \seqfz[m]{x_j}, Q = \seqfz{y_j} \in \scp([a, b])$, then $Q$ is \textbf{finer} than $P$ if for every $0 \le j \le m$, there exists $0 \le k \le m$ such that $x_j = y_k$. For any $P, Q \in \scp([a, b])$, denote $P \le Q$ if $Q$ is finer than $P$, then
\begin{enumerate}
\item $\scp([a, b])$/$\scp_t([a, b])$ equipped with $\le$ is a upward-directed set.
\item If $P \le Q$, then $\sigma(P) \ge \sigma(Q)$.
\item For any $\eps > 0$, there exists $P \in \scp([a, b])$ with $\sigma(P) < \eps$.
\end{enumerate}
If $(P, c), (Q, d) \in \scp_t([a, b])$, then $(Q, d)$ is finer than $(P, c)$ if $Q$ is finer than $P$.
\end{definition}

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\section{Riemann-Stieltjes Integrals and Functions of Bounded Variationo}
\label{section:rs-bv}
\begin{proposition}
\label{proposition:rs-bound}
Let $[a, b] \subset \real$, $E_1, E_2, H$ be locally convex spaces, and $E_1 \times E_2 \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, and $G: [a, b] \to E_2$.
Let $[\cdot]_H$ be a continuous seminorm on $H$, then there exists continuous seminorms $[\cdot]_1$ on $E_1$ and $[\cdot]_2$ on $E_2$ such that for any $f \in RS([a, b], G)$,
\[
\braks{\int_a^bf dG}_H \le \sup_{x \in [a, b]}[f]_1 \cdot [g]_{\var, 2}
\]
\end{proposition}
\begin{proof}
By \ref{proposition:tvs-convex-multilinear}, there exists continuous seminorms $[\cdot]_1$ on $E_1$ and $[\cdot]_2$ on $E_2$ such that $[xy]_H \le [x]_1[y]_2$ for all $(x, y) \in E_1 \times E_2$.
Let $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_t([a, b])$, then
\begin{align*}
[S(P, c, f, G)]_H &\le \sum_{j = 1}^n [f(c_j)[G(x_j) - G(x_{j - 1})]]_H \le \sum_{j = 1}^n [f(c_j)]_1[G(x_j) - G(x_{j - 1})]_2 \\
&\le \sup_{x \in [a, b]}[f]_1 \cdot V_{2, P}(G) \le \sup_{x \in [a, b]}[f]_1 \cdot [g]_{\var, 2}
\end{align*}
\end{proof}
\begin{proposition}
\label{proposition:rs-complete}
Let $[a, b] \subset \real$, $E_1, E_2, H$ be locally convex spaces, and $E_1 \times E_2 \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, and $G \in BV([a, b]; E_2)$.
For each continuous seminorm $\rho$ on $H$ and $f: [a, b] \to E$, define
\[
[f]_{u, \rho} = \sup_{x \in [a, b]}f(\rho)
\]
Let $\net{f} \subset RS([a, b], G)$ such that:
\begin{enumerate}
\item[(a)] $\rho(f_\alpha - f) \to 0$ for all continuous seminorm $\rho$ on $E_1$.
\item[(b)] $\lim_{\alpha \in A}\int_a^b f_\alpha dG$ exists.
\end{enumerate}
then $f \in RS([a, b], G)$ and $\int_a^b f dG = \lim_{\alpha \in A}\int_a^b f_\alpha dG$. In particular,
\begin{enumerate}
\item If $H$ is complete, then condition (a) may be omitted.
\item If $H$ is sequentially complete and $A = \nat$, then condition (b) may be omitted.
\end{enumerate}
\end{proposition}
\begin{proof}
Let $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_t([a, b])$, then
\begin{align*}
\rho\paren{S(P, c, f, G) - \lim_{\alpha \in A}\int_a^b f_\alpha dG} &\le \rho(S(P, c, f - f_\alpha, G)) \\
&+ \rho\paren{\int_a^b f_\alpha dG - \lim_{\alpha \in A}\int_a^b f_\alpha dG} \\
&+ \rho\paren{S(P, c, f_\alpha, G) - \int_a^b f_\alpha dG}
\end{align*}
Let $\rho$ be a continuous seminorm on $H$, and $[\cdot]_1$ and $[\cdot]_2$ be continuous seminorms on $E_1$ and $E_2$ such that $\rho(xy) \le [x]_1[y]_2$ for all $(x, y) \in E_1 \times E_2$.
Let $\eps > 0$, then by assumption (a) and (b), there exists $\alpha \in A$ such that:
\begin{enumerate}
\item $[f - f_\alpha]_1 < \eps/(3[G]_{\var, 2})$.
\item $\rho\paren{\int_a^b f_\alpha dG - \lim_{\alpha \in A}\int_a^b f_\alpha dG} < \eps/3$.
\end{enumerate}
Since $f_\alpha \in RS([a, b], G)$, there exists $P_0 \in \scp([a, b])$ such that if $P \ge P_0$,
\begin{enumerate}
\item[(3)] $\rho\paren{S(P, c, f_\alpha, G) - \int_a^b f_\alpha dG} < \eps/3$.
\end{enumerate}
Thus for any $(P, c) \in \scp_t([a, b])$ with $P \ge P_0$,
\[
\rho\paren{S(P, c, f, G) - \lim_{\alpha \in A}\int_a^b f_\alpha dG} < \eps
\]
\end{proof}
\begin{proposition}
\label{proposition:rs-bv-continuous}
Let $[a, b] \subset \real$, $E_1, E_2$ be locally convex spaces, $H$ be a sequentially complete locally convex space, and $E_1 \times E_2 \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map.
Let $f \in C([a, b]; E_1)$, $G \in BV([a, b]; E_2)$, then
\begin{enumerate}
\item $f \in RS([a, b], G)$.
\item For any $\seq{(P_n, t_n)} \subset \scp_t([a, b])$ with $\sigma(P_n) \to 0$,
\[
\int_a^b fdG = \limv{n}S(P_n, t_n, f, G)
\]
\end{enumerate}
\end{proposition}
\begin{proof}
Let $\rho$ be a continuous seminorm on $H$, and $[\cdot]_1$ and $[\cdot]_2$ be continuous seminorms on $E_1$ and $E_2$ such that $\rho(xy) \le [x]_1[y]_2$ for all $(x, y) \in E_1 \times E_2$.
Let $(P = \seqfz{x_j}, c = \seqf{c_j}), (Q = \seqfz[m]{y_j}, d = \seqf[m]{d_j}) \in \scp_t([a, b])$ with $Q \ge P$, then
\begin{align*}
\rho(S(P, c, f, G) - S(Q, d, f, G)) &\le \sum_{j = 1}^n \sum_{y_k \in [x_{j - 1}, x_j]}[f(c_j) - f(d_k)]_1[G(y_k) - G(y_{k - 1})]_2 \\
&\le \sup_{\begin{array}{c} x, y \in [a, b] \\ |x - y| < \sigma(P) \end{array}}[f(x) - f(y)]_1 \cdot [G]_{\var, 2}
\end{align*}
Therefore for any two $(P, c), (Q, d) \in \scp_t([a, b])$,
\[
\rho(S(P, c, f, G) - S(Q, d, f, G)) \le 2 \cdot \sup_{\begin{array}{c} x, y \in [a, b] \\ |x - y| < \max(\sigma(P), \sigma(Q)) \end{array}}[f(x) - f(y)]_1 \cdot [G]_{\var, 2}
\]
by passing through a common refinement. Since $f \in C([a, b]; E_1)$, this bound tends to $0$ as $\max(\sigma(P), \sigma(Q))$ tends to $0$, so $\angles{S(P, c, f, G)}_{(P, c) \in \scp_t([a, b])}$ is a Cauchy net.
In addition, for any $\seq{(P_n, t_n)}$ as in (2), $\limv{n}S(P_n, t_n, f, G)$ exists by sequential completeness. Since $\angles{S(P, c, f, G)}_{(P, c) \in \scp_t([a, b])}$ is Cauchy, the limit $\lim_{(P, c) \in \scp_t([a, b])}S(P, c, f, G)$ exists as well and is equal to $\limv{n}S(P_n, t_n, f, G)$.
\end{proof}

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\section{Riemann-Stieltjes Sums and Integrals}
\label{section:tvs-rs-integral}
\begin{definition}[Riemann-Stieltjes Sum, {{\cite[Section X.1]{Lang}}}]
\label{definition:rs-sum}
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
\[
S(P, c, f, G) = \sum_{j = 1}^n f(c_j)[G(x_j) - G(x_{j - 1})]
\]
is the \textbf{Riemann-Stieltjes sum} of $f$ with respect to $G$ and $(P, c)$.
\end{definition}
\begin{definition}[Riemann-Stieltjes Integral, {{\cite[Section X.1]{Lang}}}]
\label{definition:rs-integral}
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 \textbf{Riemann-Stieltjes integrable} with respect to $G$ if the limit
\[
\int_a^b f dG = \int_a^b f(t)dG(t) = \lim_{(P, c) \in \scp_t([a, b])}S(P, c, f, G)
\]
exists. In which case, $\int_a^b fdG$ is the \textbf{Riemann-Stieltjes integral} of $G$.
The set $RS([a, b], G)$ is the vector space of all \textbf{Riemann-Stieltjes integrable functions} with respect to $G$.
\end{definition}
\begin{lemma}[Summation by Parts, {{\cite[Proposition 1.4]{Lang}}}]
\label{lemma:sum-by-parts}
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
\[
S(P, c, f, G) + S(P', c', G, f) = f(b)G(b) - f(a)G(a)
\]
where $P' = \seqfz[n+1]{y_j} = [a, c_1, \cdots, c_n, b]$ and $c' = \seqf[n+1]{d_j} = [x_0, \cdots, x_n]$.
\end{lemma}
\begin{proof}
Denote $c_0 = a$ and $c_{n+1} = b$, then
\begin{align*}
S(P, c, f, G) &= \sum_{j = 1}^n f(c_j)[G(x_j) - G(x_{j - 1})]
= \sum_{j = 1}^n f(c_j)G(x_j) - \sum_{j = 1}^n f(c_j)G(x_{j - 1}) \\
&= f(c_n)G(x_n)- f(c_0)G(x_0) + \sum_{j = 1}^n f(c_{j - 1})G(x_{j-1}) - \sum_{j = 1}^n f(c_j)G(x_{j - 1}) \\
&= f(c_n)G(x_n)- f(c_0)G(x_0) - \sum_{j = 1}^n G(x_{j - 1})[f(c_j) - f(c_{j - 1})] \\
&= f(c_{n+1})G(x_n) - f(c_0)G(x_0) - \sum_{j = 1}^{n+1}G(x_{j - 1})[f(c_j) - f(c_{j - 1})] \\
&= f(b)G(b) - f(a)G(a) - S(P', c', G, f)
\end{align*}
\end{proof}
\begin{theorem}[Integration by Parts]
\label{theorem:rs-ibp}
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,
\[
\int_a^b f dG + \int_a^b G df = f(b)G(b) - f(a)G(a)
\]
\end{theorem}
\begin{proof}
Suppose that $f \in RS([a, b], G)$. Let $U \in \cn_F(0)$, 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
\[
Q_0 = [x_0, x_1, x_1, \cdots, x_n, x_n]
\]
then for any $(Q = \seqfz[m]{y_j}, d = \seqf[m]{d_j}) \in \scp_t([a, b])$ with $Q \ge Q_0$,
\[
f(b)G(b) - f(a)G(a) - \int_a^b fdG - S(Q, d, G, f) =
\int_a^b fdG - S(Q', d', G, f)
\]
by \ref{lemma:sum-by-parts}, where $d$ and $Q'$ contain $\seqfz{x_j}$. Thus $(Q', d') \ge P_0$, and $\int_a^b fdG - S(Q', d', G, f) \in U$.
\end{proof}