Jerry's Digital Garden

Bibliography
/Part 3: Functional Analysis/Chapter 11: The Riemann-Stieltjes Integral

11.4 Riemann-Stieltjes Integrals and Functions of Bounded Variation

Proposition 11.4.1. 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 $f \in RS([a, b], G)$,

\[\braks{\int_a^bf dG}_{H} \le \sup_{x \in [a, b]}[f]_{1} \cdot [g]_{\text{var}, 2}\]

Proof. By Proposition 9.2.2, 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]_{\text{var}, 2}\end{align*}
$\square$

Proposition 11.4.2. 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 $E_{1}$ and $f: [a, b] \to E_{1}$, define

\[[f]_{u, \rho}= \sup_{x \in [a, b]}\rho(f(x))\]

Let $\net{f}\subset RS([a, b], G)$ such that:

  1. For each continuous seminorm $\rho$ on $E_{1}$, $[f_{\alpha} - f]_{u, \rho}\to 0$.

  2. $\lim_{\alpha \in A}\int_{a}^{b} f_{\alpha} dG$ 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. If $H$ is complete, then condition (b) may be omitted.

  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 $[\cdot]_{1}$ 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:

  1. $[f - f_{\alpha}]_{1} < \eps/(3[G]_{\text{var}, 2})$.

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

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 11.4.3. 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

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

  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 $[\cdot]_{1}$ 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]_{\text{var}, 2}\end{align*}

Therefore for any two $(P, c), (Q, d) \in \scp_{t}([a, b])$,

\[\rho(S(P, c, f, G) - S(Q, d, f, G)) \le 2 \cdot \sup_{\begin{array}{c}x, y \in [a, b] \\ |x - y| < \max(\sigma(P), \sigma(Q))\end{array}}[f(x) - f(y)]_{1} \cdot [G]_{\text{var}, 2}\]

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

Direct References

Jerry's Digital Garden

Bibliography

Direct References