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/Part 3: Functional Analysis/Chapter 5: Riemann-Stieltjes Integral

5.4 Integrators of Bounded Variation

Proposition 5.4.1 (Integrators Inequality).label Let $[a,b]\suf\R$, $E,F,H$ be locally convex spaces, $E\ti F\to H$ with $(x,y)\mto xy$ be a continuous bilinear map, $(P=\curl{x_j}_{0}^{n},c=\curl{c_j }_{1}^{n})\in\scr{P}_{t}([a,b])$ and $G:[a,b]\to F$. Let $\braks{\cd}_{H}$ be a continuous seminorm on $H$, then there exists continuous seminorms $\braks{\cd}_{E },\braks{\cd}_{F }$ on $E,F$ respectively such that

\begin{align*}\braks{S(P,c,f,G)}_{H }\leq \SUP{x\in[a,b]}\braks{f }_{E }\cd \braks{G }_{\text{var},F }\end{align*}

In particular for any $f\in RS([a,b],G)$

\begin{align*}\braks{\integral{a}{b}fdG}_{H}\leq \SUP{x\in[a,b]}\braks{f}_{E}\cd\braks{G}_{\text{var},F}\end{align*}

Proof. By Proposition 4.2.3, there exists continuous seminorms $\braks{\cd}_{E}$ on $E$ and $\braks{\cd}_{F }$ on $F$ such that $\braks{xy}_{H }\leq \braks{x }_{E }\braks{y }_{F }$ for all $(x,y)\in E\ti F$ then

\begin{align*}\braks{S(P,c,f,G)}_{H }&\leq \sums{n }{j=1 }\braks{f(c_j)\braks{G(x_j)-G(x_{j-1})}}_{H }\\&\leq \sums{n }{j=1 }\braks{f(c_j)}_{E}\braks{\braks{G(x_j)-G(x_{j-1})}}_{F}\\&\leq \SUP{x\in[a,b]}\braks{f}_{E}\cd V_{F ,P }\parens{G }\\&\leq \SUP{x\in[a,b]}\braks{f }_{E }\cd \braks{G }_{\text{var},F }\end{align*}

Since $(P,c)$ was arbitrary we are done.$\square$

Proposition 5.4.2 (Limit and Integral).label Let $[a,b]\suf\R$, $E,F,H$ be locally convex spaces, $E\ti F\to H$ with $(x,y)\mto xy$ 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

\begin{align*}\braks{f }_{s,\rho}\define \SUP{x\in[a,b]}\rho(f(x))\end{align*}

Let $\ang{f_\al}_{\al\in A}\suf RS([a,b],G)$ such that

  1. (a)

    For each continuous seminorm $\rho$ on $E$, $\braks{f_\al-f }_{s,\rho}\to 0$

  2. (b)

    $\limit{\al\in A}\integral{a }{b }f_{\al} dG$ exists.

then $f\in RS([a,b],G)$ and $\integral{a }{b }fdG=\limit{\al\in A}\integral{a }{b }f_{\al} dG$. In particular the condition (b) may be omitted if $H$ is either complete or $H$ is sequentially complete and $A=\N^{+}$.

Proof. Let $(P=\curl{x_j}_{0}^{n},c=\curl{c_j}_{1}^{n})\in \scr{P}_{t}([a,b])$, then for any continuous seminorm $\braks{\cd}_{H }$ on $H$

\begin{align*}\braks{S(P,c,f,G)-\limit{\al\in A}\integral{a }{b }f_\al dG}_{H}&\leq \braks{S(P,c,f-f_\al,G)}_{H }\\&+\braks{S(P,c,f_\al,G)-\integral{a }{b }f_\al dG}_{H}\\&+\braks{\integral{a }{b }f_\al dG -\limit{\al\in A}\integral{a }{b }f_\al dG}_{H}\end{align*}

By Proposition 4.2.3, there exists continuous seminorms $\braks{\cd}_{E}$ on $E$ and $\braks{\cd}_{F }$ on $F$ such that $\braks{xy}_{H }\leq \braks{x }_{E }\braks{y }_{F }$ for all $(x,y)\in E\ti F$. Let $\e>0$, then by assumptions (a) and (b), there exists $\al\in A$ such that:

  1. (1)

    $\braks{f-f_\al}_{E }<\frac{\e}{3 \braks{G }_{\text{var},F }}$

  2. (2)

    $\braks{\integral{a }{b }f_\al dG-\limit{\al\in A}\integral{a }{b }f_\al dG}_{H}<\frac{\e }{3}$

  3. (3)

    Since $f_{\al}\in RS([a,b],G)$, there exists $P_{0}\in \scr{P}([a,b])$ such that for any $P\geq P_{0}$,

  4. (4)

    $\braks{S(P,c,f_\al,G)-\integral{a }{ b }f_\al dG }_{H }\frac{\e}{3}$

Thus for any $(P,c)\in \scr{P}_{t}([a,b])$ with $P\geq P_{0}$,
\begin{align*}\braks{S(P,c,f,G)-\limit{\al\in A}\integral{a }{b }f_\al dG }_{H }<\e\end{align*}

$\square$

Proposition 5.4.3 (Equicontinuous and Integral).label Let $[a,b]\suf \R$, $E,F$ be locally convex spaces, $H$ be a sequentially complete locally convex space, and $E\ti F\to H$ with $(x,y)\mto xy$ 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)

    If $\cali{F}\suf C([a,b];E)$ is equicontinuous and $\curl{(P_n,t_n)_{n\geq 1}}\suf \scr{P}_{t}([a,b])$ with $\s(P_{n})\to 0$ then

    \begin{align*}\integral{a }{b }f dG=\limit{n\to\infty}S(P_{n},t_{n},f,G)\end{align*}

    uniformly for all $f\in \cali{F}$

Proof. Fix a continuous seminorm $\braks{\cd }_{H }$ then by Proposition 4.2.3, there exists continuous seminorms $\braks{\cd}_{E}$ on $E$ and $\braks{\cd}_{F }$ on $F$ such that $\braks{xy}_{H }\leq \braks{x }_{E }\braks{y }_{F }$ for all $(x,y)\in E\ti F$. Let $(P=\curl{x_j}_{0}^{n},c=\curl{c_j }_{1}^{n}),(Q=\curl{y_j }_{0}^{m},d=\curl{d_j }^{m}_{1})\in \scr{P}_{t}([a,b])$ with $Q\geq P$, then since each interval $[x_{j-1},x_{j}]$ is subdivided by some points of $Q$ also whenever $y_{k}\in [x_{j-1},x_{j}]$ we have $c_{j}\in [x_{j-1},x_{j}],d_{k}\in [y_{k-1},y_{k}]\suf [x_{j-1},x_{j}]$ so $\abs{c_j-d_k}\leq \s(P)$ hence we have

\begin{align*}\braks{S(P,c,f,G)-S(Q,d,f,G)}_{H }&\leq \sums{n }{j=1}\sums{}{y_k\in [x_{j-1},x_j]}\braks{f(c_j)-f(d_k)}_{E}\braks{G(y_k)-G(y_{k-1})}_{F }\\&\leq \SUP{x,y\in[a,b],\abs{x-y}<\s(P)}\braks{f(x)-f(y)}_{E }\cd \braks{G}_{\text{var},F}\end{align*}

Therefore for any $(P,c),(Q,d)\in \scr{P}_{t}([a,b])$ we have

\begin{align}\braks{S(P,c,f,G)-S(Q,d,f,G)}_{H }&\leq \braks{S(P,c,f,G)-S(R,e,f,G)+S(R,e,f,G)-S(Q,d,f,G)}_{H }\\&\leq 2\cd \SUP{x,y\in[a,b],\abs{x-y}<\max \curl{\s(P),\s(Q)}}\braks{f(x)-f(y)}_{E }\cd \braks{G}_{\text{var},F}\tag{5.1}\end{align}

where $R$ is a common refine hence $\s(R)\leq\max \curl{\s(P),\s(Q)}$. Consider the filter on tagged partitions generated by the sets

\begin{align*}\cali{U}_{\de}\define \curl{(P,c)\in \scr{P}_t([a,b]):\s(P)<\de}&&\de>0\end{align*}

If $\cali{S}:(P,c)\mto S(P,c,f,G)$ then $\cali{S}(\ang{\cali{U}_\de})$ is a Cauchy filter in $H$. Indeed, since $f\in C([a,b];E)=UC([a,b];E)$ by Proposition 5.3.7, we have $\braks{S(P,c,f,G)-S(Q,d,f,G)}_{H }\to 0$ as $\max \curl{\s(P),\s(Q)}\to 0$ by 5.1. In particular, for $\curl{(P_n,t_n)}$ as in (2), $\limit{n\to \infty}S(P_{n},t_{n},f,G)$ exists by sequential completeness of $H$ and is unique by 5.1 which proves (1). Now for $\cali{F}$ as in (2) we have by equicontinuity

\begin{align*}\SUP{f\in \cali{F}}\SUP{x,y\in [a,b],\abs{x-y}<\de}\to 0,\text{ as }\de\to 0\end{align*}

so by 5.1

\begin{align*}\SUP{f\in \cali{F}}\braks{S(P,c,f,G)-S(Q,d,f,G)}_{H }&\leq 2\cd \SUP{f\in\cali{F}}\SUP{x,y\in[a,b],\abs{x-y}<\max \curl{\s(P),\s(Q)}}\braks{f(x)-f(y)}_{E }\cd \braks{G}_{\text{var},F}\end{align*}

showing $\limit{(P,c)\in\scr{P}_t([a,b])}S(P,c,f,G)$ exists and is equal to $\limit{n\to \infty}S(P_{n},t_{n},f,G)$.$\square$

Theorem 5.4.4 (Fubini’s Theorem for Riemann-Stieltjes Integrals).label Let $[a,b],[c,d]\suf\R$, $E,F,G,H$ be a locally convex space over $K\in \curl{\R,\com}$ with $H$ sequentially complete, $E\ti F\ti G\to H$ with $(x,y,z)\mto xyz$ is a 3-linear map, $\al\in BV([a,b];F),\be\in BV([c,d];G)$ and $f\in C([a,b]\ti [c,d];E)$ then

\begin{align*}\integral{a }{b }\integral{c }{d }f(s,t)\be(dt)\al(ds)=\integral{c }{d }\integral{a }{b }f(s,t)\al(ds)\be(dt)\end{align*}

Proof. Define the continuous bilinear maps

\begin{align*}\lam:E\ti F\to L(G;H)&&\lam(x,y)\curl{z}\define xyz\\ \mu:E\ti G\to L(F;H)&&\mu(x,z)\curl{y}\define xyz\end{align*}

By Proposition 5.3.7 $f\in C([a,b]\ti [c,d];E)=UC([a,b]\ti[c,d];E)$ hence $\curl{f(\cd,t):t\in[c,d]}\suf C([a,b];E)$ is uniformly equicontinuous so by Proposition 5.4.3 we can define

\begin{align*}g:[a,b]\to L(F;H)&&s\mto \integral{c }{d }\mu\parens{f(s,t),\be(dt)}\end{align*}

then observe for any $(P=\curl{x_j}_{0}^{n},c=\curl{c_j}_{1}^{n})\in \scr{P}_{t}([a,b])$

\begin{align*}S(P,c,g,\al)&=\sums{n }{j=1}g(c_{j})\curl{\al(x_j)-\al(x_{j-1})}\\&=\sums{n }{j=1}\integral{c }{d }\mu\parens{f(s,t),\be(dt)}\curl{\al(x_j)-\al(x_{j-1})}\\&=\integral{c }{d }\sums{n }{j=1}f(c_{j},t)\parens{\al(x_j)-\al(x_{j-1})}\be(dt)\\&=\integral{c }{d }S(P_{n},c_{n},\lam(f(\cd,t),\cd)\al)\be(dt)\end{align*}

Since $\al\in BV([a,b];F)$ by Proposition 5.4.3, for any $\curl{(P_n,c_n)_{n\geq 1}}\suf \scr{P}_{t}([a,b])$ with $\s(P_{n})\to 0$

\begin{align*}\integral{a }{b }\integral{c }{d }f(s,t)\be(dt)\al(ds)=\integral{a }{b }g(s)\al(ds)=\limit{n\to\infty}S(P_{n},c_{n},g,\al)\end{align*}

By Proposition 5.3.7 $f\in C([a,b]\ti [c,d];E)=UC([a,b]\ti[c,d];E)$ hence $\curl{f(\cd,t):t\in[c,d]}\suf C([a,b];E)$ is uniformly equicontinuous so by Proposition 5.4.3

\begin{align*}\limit{n\to\infty}S(P_{n},c_{n},\lam(f(\cd,t),\cd)\al)=\integral{a }{b}\lam\parens{f(s,t),\al(ds)}\end{align*}

uniformly for all $t\in [c,d]$. As $\be\in BV([c,d];G)$ we may interchange limit and integral by Proposition 5.4.2 to obtain

\begin{align*}\limit{n\to\infty}\integral{c }{d }S(P_{n},c_{n},\lam \parens{f(\cd,t),\cd},\al)\be(dt)=\integral{c }{d }\integral{a }{b }\lam \parens{f(s,t),\al(ds)}\curl{\be(dt)}=\integral{c }{d }\integral{a }{b}f(s,t)\al(ds)\be(dt)\end{align*}

so we are done by the observation.$\square$

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