Let $-\infty \le a < b \le \infty$, $E_1, E_2, H$ be [[Normed Vector Space|normed spaces]] over $F \in \bracs{\real, \complex}$, $E_1 \times E_2 \to H$ with $(x, y) \mapsto xy$ be a bounded [[Bounded Multilinear Map|bounded bilinear map]] such that $\norm{xy}_{H} \le \norm{x}_{E_1}\norm{y}_{E_2}$ for all $(x, y) \in E_1 \times E_2$, and $f: [a, b] \to E_1$, $G: [a, b] \to E_2$ be [[Function|functions]]. > [!definition] > > Let $(P = \seqf{x_j}, c = \seqf{c_j})$ be a tagged [[Partition|partition]], then > $ > S(P, c, f, G) = \sum_{j = 1}^{n}f(c_j)[G(x_j) - G(x_{j - 1})] > $ > the **Riemann-Stieltjes sum** of $f$ with respect to $G$ and $(P, c)$. The function $f$ is **Riemann-Stieltjes integrable** with respect to $F$ if the [[Net|net]] limit over all tagged partitions > $ > \int_a^b fdG = \int_a^bf(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 **Riemann-Stieltjes integral** of $f$ with respect to $G$. > > Let $RS([a, b], G)$ be the space of all functions that are Riemann-Stieltjes integrable with respect to $G$, then $RS([a, b], G)$ is a $F$-[[Vector Space|vector space]], and $f \mapsto \int_a^b fdG$ is a [[Linear Functional|linear functional]] on $RS([a, b], G)$. > > *Proof*. Let $g \in RS([a, b], G)$ and $\lambda \in F$, then > $ > \begin{align*} > S(P, c, \lambda f + g, G) &= \sum_{j = 1}^{n}[\lambda f(c_j) + g(c_j)][G(x_j) - G(x_{j - 1})] \\ > &= \lambda S(P, c, f, G) + S(P, c, g, G) > \end{align*} > $ > thus > $ > \begin{align*} > \int_a^b (\lambda f + g)dG &= \lim_{(P, c) \in \scp_t([a, b])}\braks{\lambda S(P, c, f, G) + S(P, c, g, G)} \\ > &= \lambda \int_a^bf dG + \int_a^bgdG > \end{align*} > $ > exists and $\lambda f + g \in RS([a, b], G)$. > [!theorem] Integration By Parts > > Let $(P = \seqfz{x_j}, c = \seqf{c_j})$ be a tagged partition, 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 $c' = \seqf[n + 1]{d_j} = [x_0, \cdots, x_n]$. > > Therefore $f \in RS([a, b], G)$ if and only if $g \in RS([a, b], F)$, in which case > $ > \int_a^b fdG + \int_a^b Gdf = f(b)G(b) - f(a)G(a) > $ > *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)G(x_n) - f(c_{0})G(x_0) + f(c_{n+1})G(x_{n}) - f(c_{n+1})G(x_n)\\ > &-\sum_{j = 1}^n G(x_{j - 1})[f(c_j) - f(c_{j- 1})] \\ > &= f(b)G(b) - f(a)G(a) - \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*} > $ > For integrability, let $P_G = \seqfz{x_j} \in \scp([a, b])$ such that $\normn{S(Q, d, f, G) - \int_a^b fdG}_{H} < \eps$ for all $Q \supset P_G$ and tag $d$ for $Q$. Define > $ > P_f = [x_0, x_1, x_1, x_2, x_2, \cdots, x_n, x_n] > $ > then for any $P \ge P_f$ and tag $c = \seqf[m]{c_j}$ for $P$, $\bracs{c_j: 1 \le j \le m} \supset \bracs{x_j: 1 \le j \le n}$. Therefore $(P', c') \ge (P_G, d)$ for any tag $d = \seqf{d_j}$ for $P_G$, so > $ > \begin{align*} > &\norm{f(b)G(b) - f(a)G(a) - \int_{a}^bfdG - S(P, c, G, f)}_H \\ > &= \norm{\int_a^bfdG - S(P', c',f, G)}_H < \eps > \end{align*} > $ > for all $P \ge P_f$ and tag $c$ for $P$, and > $ > \int_a^bGdf = f(b)G(b) - f(a)G(a) - \int_a^b fdG > $ # Riemann-Stieltjes Integrals and Functions of Bounded Variation In this section, assume that $G: [a, b] \to E_2$ is of [[Bounded Variation|bounded variation]]. > [!theorem] > > Let $a \le c \le d \le b$ such that $G$ is continuous at $c$ and $d$, then for any $x \in E_1$, $f(t) = x \cdot \one_{[c, d]}(t)$ is Riemann-Stieltjes integrable with respect to $G$, and > $ > \int_a^b x \cdot \one_{[c, d]}(t)dG(t) = G(d) - G(c) > $ > By applying the above to $\one_{[c, c]}$ and $\one_{[d, d]}$, we have that > $ > \int_a^b x \cdot \one_{(c, d)}dG = \int_a^b x \cdot \one_{[c, d)}dG = \int_a^b x \cdot \one_{(c, d]}dG = G(d) - G(c) > $ > the integral does not depend on the form of the interval. > > *Proof*. Let $\eps > 0$. Since $G$ is continuous at $c$ and $d$, there exists $\delta > 0$ such that $\norm{G}_{\text{var}, [c - \delta, c]}, \norm{G}_{\text{var}, [d, d + \delta]} < \eps/(2\norm{x})$. Let > $ > P = [a, c - \delta, c, d, d + \delta, b] > $ > and $(Q = \seqfz{x_j}, t = \seqf{t_j}) \in \scp_t([a, b])$ be a tagged refinement of $P$, then > $ > \begin{align*} > S(Q, t, f, G) &= \sum_{a < x_j \le c - \delta}\underbrace{f(t_j)}_{0}[G(x_j) - G(x_{j - 1})] \\ > &+\sum_{c - \delta < x_j \le c}f(t_j)[G(x_j) - G(x_{j - 1})] \\ > &+\sum_{c < x_j \le d}\underbrace{f(t_j)}_{x}[G(x_j) - G(x_{j - 1})] \\ > &+\sum_{d < x_j \le d + \delta}f(t_j)[G(x_j) - G(x_{j - 1})] \\ > &+\sum_{d + \eps < x_j \le b}\underbrace{f(t_j)}_{0}[G(x_j) - G(x_{j - 1})] \\ > &= G(d) - G(c) + \sum_{c - \delta < x_j \le c}f(t_j)[G(x_j) - G(x_{j - 1})] \\ > &+ \sum_{d < x_j \le d + \delta}f(t_j)[G(x_j) - G(x_{j - 1})] > \end{align*} > $ > Now, since $\norm{f}_u = \norm{x}_{E_1}$, > $ > \norm{\sum_{c - \delta < x_j \le c}f(t_j)[G(x_j) - G(x_{j - 1})]}_H < \norm{x}_{E_1}\norm{G}_{\text{var},[c - \eps, \eps]} < \eps/2 > $ > and > $ > \norm{\sum_{d < x_j \le d + \delta}f(t_j)[G(x_j) - G(x_{j - 1})]}_H \le \norm{x}_{E_1}\norm{G}_{\text{var}, [d , d + \eps]} < \eps/2 > $ > Thus $\norm{S(Q, t, f, G) - G(d) - G(c)}_H < \eps$. > [!theorem] > > Let $f \in RS([a, b], G)$, then > $ > \norm{\int_a^b fdG}_H \le \norm{f}_{u} \cdot \norm{G}_{\text{var}} > $ > Moreover, for any $\seq{f_n} \subset RS([a, b], G)$ and $f: [a, b] \to E_1$ such that $f_n \to f$ [[Uniform Convergence|uniformly]] and $\limv{n}\int_a^bf_ndG$ exists, $f \in RS([a, b], G)$ and $\int_a^b fdG = \limv{n}\int_a^b f_ndG$. In particular, if $H$ is a [[Banach Space|Banach space]], then $RS([a, b], G)$ is closed under uniform limits. > > *Proof*. For any tagged partition $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_t([a, b])$, > $ > \begin{align*} > \norm{S(P, c, f, G)}_H &\le \sum_{j = 1}^n \norm{f(c_j)[G(x_{j}) - G(x_{j - 1})]}_H \\ > &\le \norm{f}_u \sum_{j = 1}^n \norm{G(x_j) - G(x_{j - 1})}_{E_2} \le \norm{f}_u \cdot \norm{G}_{\text{var}} > \end{align*} > $ > Let $L = \limv{n}\int_a^b f_ndG$ and $\eps > 0$, then there exists $N \in \nat$ such that $\norm{f_n - f}_u < \eps/(3\norm{G}_{\text{var}})$ and $\normn{L - \int_a^b f_ndG}_H < \eps/3$. Let $P \in \scp([a, b])$ such that for any $(Q, c) \in \scp_t([a, b])$ with $Q \ge P$, $\normn{S(Q, c, f, G) - \int_a^b f_ndG}_N < \eps/3$. In which case, > $ > \begin{align*} > \norm{S(Q, c, f, G) - L}_H &\le \norm{S(Q, c, f - f_n, G)}_H + \norm{\int_a^b f_n dG - L}_H \\ > & + \norm{S(Q, c, f_n, G) - \int_a^bf_ndG}_H \\ > &< \norm{f - f_n}_u\norm{G}_{\text{var}} + \eps/3 + \eps/3 = \eps > \end{align*} > $ > Therefore $\int_a^b fdG = \lim_{(Q, c) \in \scp_t([a,b])} = \limv{n}\int_a^b f_n dG$. > > If $H$ is a Banach space, then $\limv{n}\int_a^b f_ndG$ always exists. Thus $RS([a, b], G)$ is closed under uniform limits. > [!theorem] > > Suppose that $H$ is a [[Banach Space|Banach space]], then all [[Continuity|continuous]] functions $[a, b] \to E_1$ are Riemann-Stieltjes integrable with respect to $G$. Moreover, for any $f \in C([a, b]; E_1)$ and tagged partitions $\seq{(P_n, t_n)}$ such that $\sigma(P_n) \to 0$[^1], > $ > \int_a^b fdG = \limv{n}S(P_n, t_n, f, G) > $ > > > *Proof*. Let $\eps > 0$, then there exists $\delta > 0$ such that $\norm{f(x) - f(y)}_{E_1} < \eps/(2\norm{G}_{\text{var}})$ whenever $\abs{x - y} < \delta$. In which case, for any $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_t([a, b])$ with $\sigma(P) < \delta$ and $(Q = \seqfz[m]{y_j}, d = \seqf[m]{d_j}) \ge (P, c)$, > $ > \begin{align*} > &\norm{S(P, c, f, G) - S(Q, d, f, G)}_H \\ > &\le \sum_{j = 1}^{n}\sum_{x_{j- 1} < y_k \le x_j}\norm{f(c_j) - f(d_k)}_{E_1}\norm{G(y_k) - G(y_{k - 1})}_{E_2} \\ > &< \frac{\eps}{2\norm{G}_{\text{var}}}\sum_{k = 1}^{m}\norm{G(y_k) - G(y_{k - 1})}_{E_2} \le \eps/2 > \end{align*} > $ > Thus the net $\angles{S(P, c, f, G)}_{(P, c) \in \scp_t([a, b])}$ is Cauchy. Since $H$ is a Banach space, there exists $L \in H$ such that $\int_a^b fdG = \lim_{(P, c) \in \scp_t([a, b])}S(P, c, f, G) = L$. > > Since any two partitions share a common refinement, if $(P', c') \in \scp_t([a, b])$ is another tagged partition with $\sigma(P') < \delta$, > $ > \norm{S(P, c, f, G) - S(P', c', f, G)}_H < \eps > $ > Therefore any sequence $\seq{(P_n, t_n)}$ where $\sigma(P_n) \to 0$ as $n \to \infty$ is [[Cauchy Sequence|Cauchy]], and there exists $L' \in H$ such that $S(P_n, t_n, f, G) \to L'$. In addition, if $\sigma(P_n) < \delta$ and $\norm{S(P_n, t_n, f, G) - L'}_H < \eps/2$, then for any tagged refinement $(Q, d) \ge (P_n, t_n)$, > $ > \normn{L' - S(Q, d, f, G)}_H \le \norm{S(P_n, t_n, f, G) - L'}_H -\norm{S(Q, d, f, G)}_H < \eps > $ > Thus $\norm{L' - L} \le \eps$. As this holds for all $\eps > 0$, $L' = L$. [^1]: For fixed $f$, such a sequential approximation can always be found. The significance of this part is to provide a condition for constructing sequences of partitions that result in convergent power sums, that applies to all continuous integrands.