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/Part 3: Functional Analysis/Chapter 13: The Riemann-Stieltjes Integral

13.4 Riemann-Stieltjes Integrals and Functions of Bounded Variation

Proposition 13.4.1.label 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 $[\cdot]_{E}$ on $E$ and $[\cdot]_{F}$ on $F$ such that for any $f \in RS([a, b], G)$,

\[\braks{\int_a^bf dG}_{H} \le \sup_{x \in [a, b]}[f]_{E} \cdot [g]_{\text{var}, F}\]

Proof. By Proposition 11.2.2, 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})]_{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*}

$\square$

Proposition 13.4.2.label 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 $E$ 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:

  1. (a)

    For each continuous seminorm $\rho$ on $E$, $[f_{\alpha} - f]_{u, \rho}\to 0$.

  2. (b)

    $\lim_{\alpha \in A}\int_{a}^{b} f_{\alpha} dG$ exists.

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,

  1. (1)

    If $H$ is complete, then condition (b) may be omitted.

  2. (2)

    If $H$ is sequentially complete and $A = \nat^{+}$, then condition (b) may be omitted.

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 $[\cdot]_{E}$ 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:

  1. (1)

    $[f - f_{\alpha}]_{E} < \eps/(3[G]_{\text{var}, F})$.

  2. (2)

    $\rho\paren{\int_a^b f_\alpha dG - \lim_{\alpha \in A}\int_a^b f_\alpha dG}< \eps/3$.

Since $f_{\alpha} \in RS([a, b], G)$, there exists $P_{0} \in \scp([a, b])$ such that if $P \ge P_{0}$,

  1. (3)

    $\rho\paren{S(P, c, f_\alpha, G) - \int_a^b f_\alpha dG}< \eps/3$.

Thus for any $(P, c) \in \scp_{t}([a, b])$ with $P \ge P_{0}$,

\[\rho\paren{S(P, c, f, G) - \lim_{\alpha \in A}\int_a^b f_\alpha dG}< \eps\]

$\square$

Proposition 13.4.3.label 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 $f \in C([a, b]; E)$, $G \in BV([a, b]; F)$, then

  1. (1)

    $f \in RS([a, b], G)$.

  2. (2)

    For any $\seq{(P_n, t_n)}\subset \scp_{t}([a, b])$ with $\sigma(P_{n}) \to 0$,

    \[\int_{a}^{b} fdG = \limv{n}S(P_{n}, t_{n}, f, G)\]

Proof. 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})]_{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 $(P, c), (Q, d) \in \scp_{t}([a, b])$,

\[\rho(S(P, c, f, G) - S(Q, d, f, G)) \le 2 \cdot \sup_{\begin{array}{c}x, y \in [a, b] \\ |x - y| < \max(\sigma(P), \sigma(Q))\end{array}}[f(x) - f(y)]_{E} \cdot [G]_{\text{var}, F}\]

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 $\limv{n}S(P_{n}, t_{n}, f, G)$.$\square$

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