From 50c231f543299dc044e28dff59044d390820b488 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Sat, 21 Mar 2026 19:15:40 -0400 Subject: [PATCH] Updated the dyadic rational numbers and RS integrals. --- src/cat/tricks/dyadic.tex | 8 ++++++- src/fa/rs/rs-bv.tex | 38 ++++++++++++++++--------------- src/fa/rs/rs-measure.tex | 23 +++++++++++++++++++ src/fa/rs/rs.tex | 48 +++++++++++++++++++++++++++++++-------- 4 files changed, 88 insertions(+), 29 deletions(-) create mode 100644 src/fa/rs/rs-measure.tex diff --git a/src/cat/tricks/dyadic.tex b/src/cat/tricks/dyadic.tex index 0cacad9..190cc0b 100644 --- a/src/cat/tricks/dyadic.tex +++ b/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 $x \le y$, $\phi(x) \le \phi(y)$. diff --git a/src/fa/rs/rs-bv.tex b/src/fa/rs/rs-bv.tex index 8bd8642..05902f8 100644 --- a/src/fa/rs/rs-bv.tex +++ b/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 \\ - &\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} + [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)]_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 \\ - &\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} + &\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)]_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} diff --git a/src/fa/rs/rs-measure.tex b/src/fa/rs/rs-measure.tex new file mode 100644 index 0000000..e8fed0e --- /dev/null +++ b/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} + diff --git a/src/fa/rs/rs.tex b/src/fa/rs/rs.tex index f05c64c..8225efe 100644 --- a/src/fa/rs/rs.tex +++ b/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} +