Updated the dyadic rational numbers and RS integrals.

src/cat/tricks/dyadic.tex

@@ -38,7 +38,13 @@
 
 \begin{proposition}
 \label{proposition:dyadic-semigroup-order}
-    Let $G$ be a commutative ordered semigroup, and $\seq{g_n} \subset G$ such that for each $n \in \natp$, $g_{n+1} + g_{n+1} \le g_n$. For each $x \in \mathbb{D} \cap [0, 1)$, let $\phi(x) = \sum_{n \in M(x)}g_n$, then
+    Let $G$ be a commutative ordered semigroup, and $\seq{g_n} \subset G$ such that 
+    \begin{enumerate}
+        \item[(a)] for each $n \in \natp$, $g_{n+1} + g_{n+1} \le g_n$.
+        \item[(b)] For each $x, y \in G$, $x + y \ge x, y$. 
+    \end{enumerate}
+    
+    For each $x \in \mathbb{D} \cap [0, 1)$, let $\phi(x) = \sum_{n \in M(x)}g_n$, then
     \begin{enumerate}
         \item For any $x, y \in \mathbb{D} \cap [0, 1)$ such that $x + y \in [0, 1)$, $\phi(x) + \phi(y) \le \phi(x + y)$.
         \item For any $x, y \in \mathbb{D} \cap [0, 1)$ with $x \le y$, $\phi(x) \le \phi(y)$.

src/fa/rs/rs-bv.tex

@@ -3,36 +3,37 @@
 
 \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 $[a, b] \subset \real$, $E, F, H$ be locally convex spaces, and $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, and $G: [a, b] \to F$.
 
-    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 $f \in RS([a, b], G)$,
+    Let $[\cdot]_H$ be a continuous seminorm on $H$, then there exists continuous seminorms $[\cdot]_E$ on $E$ and $[\cdot]_F$ on $F$ such that for any $f \in RS([a, b], G)$,
     \[
-    \braks{\int_a^bf dG}_H \le \sup_{x \in [a, b]}[f]_1 \cdot [g]_{\text{var}, 2}
+    \braks{\int_a^bf dG}_H \le \sup_{x \in [a, b]}[f]_E \cdot [g]_{\text{var}, F}
     \]
 
 \end{proposition}
 \begin{proof}
-    By \autoref{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 $(x, y) \in E_1 \times E_2$.
+    By \autoref{proposition:tvs-convex-multilinear}, there exists continuous seminorms $[\cdot]_E$ on $E$ and $[\cdot]_F$ on $F$ such that $[xy]_H \le [x]_E[y]_F$ for all $(x, y) \in E \times F$.
 
     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 \\
+        [S(P, c, f, G)]_H &\le \sum_{j = 1}^n [f(c_j)[G(x_j) - G(x_{j - 1})]]_H \\
-        &\le \sup_{x \in [a, b]}[f]_1 \cdot V_{2, P}(G) \le \sup_{x \in [a, b]}[f]_1 \cdot [g]_{\text{var}, 2}
+        &\le \sum_{j = 1}^n [f(c_j)]_E[G(x_j) - G(x_{j - 1})]_F \\
+        &\le \sup_{x \in [a, b]}[f]_E \cdot V_{2, P}(G) \le \sup_{x \in [a, b]}[f]_E \cdot [g]_{\text{var}, F}
     \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 $G \in BV([a, b]; E_2)$.
+    Let $[a, b] \subset \real$, $E, F, H$ be locally convex spaces, and $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, and $G \in BV([a, b]; F)$.
 
-    For each continuous seminorm $\rho$ on $E_1$ and $f: [a, b] \to E_1$, define
+    For each continuous seminorm $\rho$ on $E$ and $f: [a, b] \to E$, define
     \[
     [f]_{u, \rho} = \sup_{x \in [a, b]}\rho(f(x))
     \]
 
     Let $\net{f} \subset RS([a, b], G)$ such that:
     \begin{enumerate}
-        \item[(a)] For each continuous seminorm $\rho$ on $E_1$, $[f_\alpha - f]_{u, \rho} \to 0$.
+        \item[(a)] For each continuous seminorm $\rho$ on $E$, $[f_\alpha - f]_{u, \rho} \to 0$.
         \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,
@@ -48,11 +49,11 @@
         &+ \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 $\rho$ be a continuous seminorm on $H$, and $[\cdot]_E$ and $[\cdot]_F$ be continuous seminorms on $E$ and $F$ such that $\rho(xy) \le [x]_E[y]_F$ for all $(x, y) \in E \times F$.
 
     Let $\eps > 0$, then by assumption (a) and (b), there exists $\alpha \in A$ such that:
     \begin{enumerate}
-        \item $[f - f_\alpha]_1 < \eps/(3[G]_{\text{var}, 2})$.
+        \item $[f - f_\alpha]_E < \eps/(3[G]_{\text{var}, F})$.
         \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 $P \ge P_0$,
@@ -68,9 +69,9 @@
 
 \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 $[a, b] \subset \real$, $E, F$ be locally convex spaces, $H$ be a sequentially complete locally convex space, and $E \times F \to H$ 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
+    Let $f \in C([a, b]; E)$, $G \in BV([a, b]; F)$, then
     \begin{enumerate}
         \item $f \in RS([a, b], G)$.
         \item For any $\seq{(P_n, t_n)} \subset \scp_t([a, b])$ with $\sigma(P_n) \to 0$,
@@ -81,19 +82,20 @@
     \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 $\rho$ be a continuous seminorm on $H$, and $[\cdot]_E$ and $[\cdot]_F$ be continuous seminorms on $E$ and $F$ such that $\rho(xy) \le [x]_E[y]_F$ for all $(x, y) \in E \times F$.
 
     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 \\
+        &\rho(S(P, c, f, G) - S(Q, d, f, G)) \\\
-        &\le \sup_{\begin{array}{c} x, y \in [a, b] \\ |x - y| < \sigma(P) \end{array}}[f(x) - f(y)]_1 \cdot [G]_{\text{var}, 2}
+        &\le \sum_{j = 1}^n \sum_{y_k \in [x_{j - 1}, x_j]}[f(c_j) - f(d_k)]_E[G(y_k) - G(y_{k - 1})]_F \\
+        &\le \sup_{\begin{array}{c} x, y \in [a, b] \\ |x - y| < \sigma(P) \end{array}}[f(x) - f(y)]_E \cdot [G]_{\text{var}, F}
     \end{align*}
     Therefore for any two $(P, c), (Q, d) \in \scp_t([a, b])$,
     \[
-    \rho(S(P, c, f, G) - S(Q, d, f, G)) \le 2 \cdot \sup_{\begin{array}{c} x, y \in [a, b] \\ |x - y| < \max(\sigma(P), \sigma(Q)) \end{array}}[f(x) - f(y)]_1 \cdot [G]_{\text{var}, 2}
+    \rho(S(P, c, f, G) - S(Q, d, f, G)) \le 2 \cdot \sup_{\begin{array}{c} x, y \in [a, b] \\ |x - y| < \max(\sigma(P), \sigma(Q)) \end{array}}[f(x) - f(y)]_E \cdot [G]_{\text{var}, F}
     \]
 
-    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.
+    by passing through a common refinement. Since $f \in C([a, b]; E)$, 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}

src/fa/rs/rs-measure.tex

@@ -0,0 +1,23 @@
+\section{The Lebesgue-Stieltjes Integral}
+\label{section:riemann-lebesgue-stieltjes}
+
+\begin{theorem}
+\label{theorem:riemann-lebesgue-stieltjes}
+    Let $[a, b] \subset \real$, $E$ be a Banach space, and $G \in BV([a, b]; E^*)$, then there exists a unique $\mu_G \in M_R([a, b]; E^*)$ such that:
+    \begin{enumerate}
+        \item For any $f \in C([a, b]; E)$, 
+        \[
+        \int f(t) G(dt) = \int \dpn{f(t), \mu_G(dt)}{E}
+        \]
+
+        \item For any 
+
+        \item For any $a \le c < d \le b$ such that $G$ is continuous on $c$ and $d$, $\mu_G((c, d]) = G(d) - G(c)$.
+    \end{enumerate}
+\end{theorem}
+\begin{proof}
+    (1): By \autoref{proposition:rs-bv-continuous}, the mapping $f \mapsto \int f(t)G(dt)$ is a continuous linear functional on $C_0([a, b]; E) = C([a, b]; E)$. Thus (1) holds by \hyperref[Singer's Representation Theorem]{theorem:singer-representation}.
+
+    (2): 
+\end{proof}
+

src/fa/rs/rs.tex

@@ -3,9 +3,9 @@
 
 \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 $[a, b] \subset \real$, $E, F, H$ 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 $f: [a, b] \to E_1$ and $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_t([a, b])$, then
+    Let $f: [a, b] \to E$ and $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_t([a, b])$, then
     \[
     S(P, c, f, G) = \sum_{j = 1}^n f(c_j)[G(x_j) - G(x_{j - 1})]
     \]
@@ -15,9 +15,9 @@
 
 \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 $[a, b] \subset \real$, $E, F, H$ 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 $f: [a, b] \to E_2$, then $f$ is \textbf{Riemann-Stieltjes integrable} with respect to $G$ if the limit
+    Let $f: [a, b] \to F$, then $f$ is \textbf{Riemann-Stieltjes integrable} with respect to $G$ if the limit
     \[
     \int_a^b f dG = \int_a^b f(t)G(dt) = \lim_{(P, c) \in \scp_t([a, b])}S(P, c, f, G)
     \]
@@ -29,7 +29,7 @@
 
 \begin{lemma}[Summation by Parts]
 \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
+    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
     \[
     S(P, c, f, G) + S(P', c', G, f) = f(b)G(b) - f(a)G(a)
     \]
@@ -40,8 +40,10 @@
     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}) \\
+        &= \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)
@@ -50,16 +52,16 @@
 
 \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 $[a, b] \subset \real$, $E, F, H$ be TVSs over $K \in \RC$, and $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map.
 
-    Let $f: [a, b] \to E_1$ 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,
+    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,
     \[
     \int_a^b f dG + \int_a^b G df = f(b)G(b) - f(a)G(a)
     \]
 
 \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
+    Suppose that $f \in RS([a, b], G)$. Let $U \in \cn_K(0)$, 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
     \[
     Q_0 = [x_0, x_1, x_1, \cdots, x_n, x_n]
     \]
@@ -72,3 +74,29 @@
 
     by \autoref{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}
+
+\begin{proposition}
+\label{proposition:rs-interval}
+    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 linear map. 
+    
+    Let $G: [a, b] \to F$ and $[c, d] \subset [a, b]$ such that $G$ is continuous at $c$ and $d$, then for any $x \in E$, $x \cdot \one_{[c, d]} \in RS([a, b], G)$, and
+    \[
+    \int_a^b x \cdot \one_{[c, d]} dG = x \cdot [G(d) - G(c)]
+    \]
+\end{proposition}
+\begin{proof}
+    Assume without loss of generality that $a < c \le d < b$. Let $U \in \cn_H(0)$, then there exists $V \in \cn_F(0)$ such that $xV \subset U$. By continuity of $G$, there exists $\delta > 0$ such that $G((c - \delta, c]) - G(c) \subset V$ and $G([d, d + \delta)) - G(d) \subset V$. In which case, for any tagged partition $(P = \bracsn{x_j}_0^n, t = \seqf{t_j})$ that contains $\bracs{c - \delta, c, d, d + \delta}$, 
+    \begin{align*}
+        S(Q, t, x \cdot \one_{[c, d]}, G) &= x\sum_{a < x_j \le c - \delta}\one_{[c, d]}(t_j)[G(x_j) - G(x_{j - 1})] \\
+        &+ x \sum_{c - \delta < x_j \le c}\one_{[c, d]}(t_j)[G(x_j) - G(x_{j - 1})] \\
+        &+ x \sum_{c < x_j \le d}\one_{[c, d]}(t_j)[G(x_j) - G(x_{j-1})] \\
+        &+ x \sum_{d < x_j \le d + \delta}\one_{[c, d]}(t_j)[G(x_j) - G(x_{j - 1})] \\
+        &+ x \sum_{d + \delta < x_j \le b}\one_{[c, d]}(t_j)[G(x_j) - G(x_{j - 1})] \\
+        &= x \sum_{c - \delta \le x_j \le c}\one_{[c, d]}(t_j)[G(x_j) - G(x_{j - 1})] \\
+        &+ x \cdot [G(d) - G(c)] \\
+        &+ x \sum_{d < x_j \le d + \delta}\one_{[c, d]}[G(x_j) - G(x_{j - 1})] \\
+        &\in G(d) - G(c) + xG([c - \delta, c]) + xG([d, d + \delta]) \\
+        &\subset x \cdot [G(d) - G(c)] + xV + xV \subset [G(d) - G(c)] + U + U
+    \end{align*}
+\end{proof}
+