Skip to main content ☰ Contents You! < Prev ^ Up Next > \(\def\ann{\operatorname{ann}}
\newcommand{\Ass}{\operatorname{Ass}}
\def\Aut{\operatorname{Aut}}
\def\can{{\mathrm {can}}}
\def\char{\operatorname{char}}
\def\cp{\operatorname{CharPoly}}
\def\codim{\operatorname{codim}}
\def\coker{\operatorname{coker}}
\DeclareMathOperator*{\colim}{colim}
\def\cont{\operatorname{cont}}
\def\diam{\operatorname{diam}}
\def\dm{\operatorname{dim}}
\DeclareMathOperator{\edim}{embdim}
\def\End{\operatorname{End}}
\def\eval{\operatorname{eval}}
\def\Ext{\operatorname{Ext}}
\def\Frac{\operatorname{Frac}}
\def\Fun{\operatorname{Fun}}
\def\Gal{\operatorname{Gal}}
\def\gcd{\operatorname{gcd}}
\newcommand{\GL}{\operatorname{GL}}
\newcommand{\ht}{\operatorname{height}}
\def\Hom{\operatorname{Hom}}
\def\id{\operatorname{id}}
\def\im{\operatorname{im}}
\def\Inn{\operatorname{Inn}}
\def\ker{\operatorname{ker}}
\def\lcm{\operatorname{lcm}}
\def\Mat{\operatorname{Mat}}
\newcommand{\Min}{\operatorname{Min}}
\def\mp{\operatorname{MinPoly}}
\def\mSpec{\operatorname{mSpec}}
\def\MSpec{\operatorname{MSpec}}
\def\null{\operatorname{Nul}}
\DeclareMathOperator{\ns}{nullspace}
\newcommand{\opp}{\operatorname{opp}}
\def\Orb{\operatorname{Orb}}
\def\Out{\operatorname{Out}}
\def\Perm{\operatorname{Perm}}
\def\ptstab{\operatorname{PtStab}}
\def\rad{\operatorname{rad}}
\DeclareMathOperator{\range}{range}
\def\rank{\operatorname{rank}}
\def\res{\operatorname{res}}
\def\setstab{\operatorname{SetStab}}
\def\sign{{\operatorname{sign}}}
\newcommand{\SL}{\operatorname{SL}}
\def\Span{\operatorname{Span}}
\def\Spec{\operatorname{Spec}}
\def\Stab{\operatorname{Stab}}
\DeclareMathOperator{\Supp}{Supp}
\def\Syl{\operatorname{Syl}}
\def\Tor{\operatorname{Tor}}
\def\trace{\operatorname{trace}}
\def\uSpec{\operatorname{\underline{Spec}}}
\newcommand{\Ob}{\mathrm{Ob}}
\newcommand{\Set}{\mathbf{Set}}
\newcommand{\Grp}{\mathbf{Grp}}
\newcommand{\Ab}{\mathbf{Ab}}
\newcommand{\Sgrp}{\mathbf{Sgrp}}
\newcommand{\Ring}{\mathbf{Ring}}
\newcommand{\Fld}{\mathbf{Fld}}
\newcommand{\cRing}{\mathbf{cRing}}
\newcommand{\Mod}[1]{#1-\mathbf{Mod}}
\newcommand{\Cx}[1]{#1-\mathbf{Comp}}
\newcommand{\vs}[1]{#1-\mathbf{vect}}
\newcommand{\Vs}[1]{#1-\mathbf{Vect}}
\newcommand{\vsp}[1]{#1-\mathbf{vect}^+}
\newcommand{\Top}{\mathbf{Top}}
\newcommand{\Setp}{\mathbf{Set}_*}
\newcommand{\Alg}[1]{#1-\mathbf{Alg}}
\newcommand{\cAlg}[1]{#1-\mathbf{cAlg}}
\newcommand{\PO}{\mathbf{PO}}
\newcommand{\Cont}{\mathrm{Cont}}
\newcommand{\MaT}[1]{\mathbf{Mat}_{#1}}
\newcommand{\Rep}[2]{\mathbf{Rep}_{#1}(#2)}
\def\l{\lambda}
\def\lx{\lambda_x}
\newcommand{\a}{\alpha}
\def\b{\beta}
\def\d{\delta}
\def\e{\varepsilon}
\def\g{\gamma}
\def\t{\theta}
\def\s{\sigma}
\def\z{\zeta}
\def\vp{\varphi}
\newcommand{\A}{\mathbb{A}}
\newcommand{\B}{\mathbb{B}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\D}{\mathbb{D}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\G}{\mathbb{G}}
\newcommand{\H}{\mathbb{H}}
\newcommand{\I}{\mathbb{I}}
\newcommand{\J}{\mathbb{J}}
\newcommand{\K}{\mathbb{K}}
\newcommand{\L}{\mathbb{L}}
\newcommand{\M}{\mathbb{M}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\O}{\mathbb{O}}
\newcommand{\P}{\mathbb{P}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\S}{\mathbb{S}}
\newcommand{\T}{\mathbb{T}}
\newcommand{\U}{\mathbb{U}}
\newcommand{\V}{\mathbb{V}}
\newcommand{\W}{\mathbb{W}}
\newcommand{\X}{\mathbb{X}}
\newcommand{\Y}{\mathbb{Y}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\ON}{\mathbb{ON}}
\def\cA{\mathcal A}
\def\cB{\mathcal B}
\def\cC{\mathcal C}
\def\cD{\mathcal D}
\def\cE{\mathcal E}
\def\cF{\mathcal F}
\def\cG{\mathcal G}
\def\cH{\mathcal H}
\def\cI{\mathcal I}
\def\cJ{\mathcal J}
\def\cK{\mathcal K}
\def\cL{\mathcal L}
\def\cM{\mathcal M}
\def\cN{\mathcal N}
\def\cO{\mathcal O}
\def\cP{\mathcal P}
\def\cQ{\mathcal Q}
\def\cR{\mathcal R}
\def\cS{\mathcal S}
\def\cT{\mathcal T}
\def\cU{\mathcal U}
\def\cV{\mathcal V}
\def\cW{\mathcal W}
\def\cX{\mathcal X}
\def\cY{\mathcal Y}
\def\cZ{\mathcal Z}
\newcommand{\fa}{{\mathfrak a}}
\newcommand{\fb}{{\mathfrak b}}
\newcommand{\fc}{{\mathfrak c}}
\newcommand{\fd}{{\mathfrak d}}
\newcommand{\fe}{{\mathfrak e}}
\newcommand{\fm}{{\mathfrak m}}
\newcommand{\fp}{{\mathfrak p}}
\newcommand{\fq}{{\mathfrak q}}
\newcommand{\fA}{{\mathfrak A}}
\newcommand{\fB}{{\mathfrak B}}
\newcommand{\fC}{{\mathfrak C}}
\newcommand{\fD}{{\mathfrak D}}
\newcommand{\fE}{{\mathfrak E}}
\newcommand{\fF}{{\mathfrak F}}
\newcommand{\fG}{{\mathfrak G}}
\newcommand{\fH}{{\mathfrak H}}
\newcommand{\fK}{{\mathfrak K}}
\newcommand{\fR}{{\mathfrak R}}
\def\sA{\mathscr A}
\def\sB{\mathscr B}
\def\sC{\mathscr C}
\def\sD{\mathscr D}
\def\sE{\mathscr E}
\def\sF{\mathscr F}
\def\sG{\mathscr G}
\def\sH{\mathscr H}
\def\sI{\mathscr I}
\def\sJ{\mathscr J}
\def\sK{\mathscr K}
\def\sL{\mathscr L}
\def\sM{\mathscr M}
\def\sN{\mathscr N}
\def\sO{\mathscr O}
\def\sP{\mathscr P}
\def\sQ{\mathscr Q}
\def\sR{\mathscr R}
\def\sS{\mathscr S}
\def\sT{\mathscr T}
\def\sU{\mathscr U}
\def\sV{\mathscr V}
\def\sW{\mathscr W}
\def\sX{\mathscr X}
\def\sY{\mathscr Y}
\def\sZ{\mathscr Z}
\def\tS{\tilde{S}}
\def\sdp{\rtimes}
\newcommand{\tensor}{\otimes}
\newcommand{\igen}[1]{\langle #1 \rangle}
\def\nsg{\unlhd}
\def\kval{{k-\mathrm{valued}}}
\def\kalg{{k-\mathrm{alg}}}
\newcommand\GG[2]{\Gal(#1/#2)}
\newcommand{\MF}[3]{\Mat_{#1\times #2}(#3)}
\newcommand{\vectwo}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}}
\newcommand{\vecthree}[3]{\begin{bmatrix} #1 \\ #2 \\ #3\end{bmatrix}}
\def\ob{{\mathfrak{ob}} }
\def\qed{\square}
\def\sse{\subseteq}
\def\ss{\subset}
\def\ssne{\subsetneq}
\def\sm{\setminus}
\def\inv{^{-1}}
\newcommand{\es}{\emptyset}
\newcommand{\Zm}[1]{\Z/({#1})}
\def\ov#1{\overline{#1}}
\def\xdots{x_1, \dots, x_n}
\def\adots{a_1, \dots, a_n}
\def\bdots{b_1, \dots, b_n}
\def\udots{u_1, \dots, u_n}
\newcommand{\leg}[2]{\left(\frac{{#1}}{{#2}}\right)}
\def\th{^{th}}
\def\htpy{\simeq_{\mathrm{htpc}}}
\def\textand{ \, \text{and} \, }
\def\textor{ \, \text{or} \, }
\def\textfor{ \, \text{for} \, }
\def\textfa{ \, \text{for all} \, }
\def\textst{ \, \text{such that} \, }
\def\textin{ \, \text{in} \, }
\def\fg{ \, \text{finitely generated} \, }
\newcommand{\op}{\mathrm{op}}
\newcommand{\xra}[1]{\xrightarrow{#1}}
\newcommand{\xora}[1]{\xtwoheadrightarrow{#1}}
\newcommand{\xira}[1]{\xhookrightarrow{#1}}
\newcommand{\xla}[1]{\xleftarrow{#1}}
\def\lra{\longrightarrow}
\def\into{\hookrightarrow}
\def\onto{\twoheadrightarrow}
\newcommand{\vv}[1]{\mathbf{#1}}
\newcommand{\lm}[2]{{#1}\,\l + {#2}\,\mu}
\renewcommand{\v}{\vv{v}}
\renewcommand{\u}{\vv{u}}
\newcommand{\w}{\vv{w}}
\newcommand{\x}{\vv{x}}
\renewcommand{\k}{\vv{k}}
\newcommand{\0}{\vv{0}}
\newcommand{\1}{\vv{1}}
\newcommand{\vecs}[2]{#1_1,#1_2,\dots,#1_{#2}}
\newcommand{\us}[1][n]{\vecs{\u}{#1}}
\newcommand{\vs}[1][n]{\vecs{\v}{#1}}
\newcommand{\ws}[1][n]{\vecs{\w}{#1}}
\newcommand{\vps}[1][n']{\vecs{\v'}{#1}}
\newcommand{\ls}[1][n]{\vecs{\l}{#1}}
\newcommand{\mus}[1][n]{\vecs{\mu}{#1}}
\newcommand{\lps}[1][n]{\vecs{\l'}{#1}}
\def\td{\tilde{\delta}}
\def\oo{\overline{\omega}}
\def\ctJ{\tilde{\mathcal J}}
\def\tPhi{\tilde{\Phi}}
\def\te{\tilde{e}}
\def\M{\operatorname{M}}
\newcommand{\homotopic}{\simeq}
\newcommand{\homeq}{\cong}
\newcommand{\iso}{\approx}
\newcommand{\dual}{\vee}
\DeclarePairedDelimiter{\abs}{|}{|}
\newcommand{\bv}{{\bar{v}}}
\newcommand{\bu}{{\bar{u}}}
\newcommand{\bw}{{\bar{w}}}
\newcommand{\by}{{\bar{y}}}
\newcommand{\ba}{{\bar{a}}}
\newcommand{\bb}{{\bar{b}}}
\newcommand{\bx}{{\bar{x}}}
\DeclarePairedDelimiterX\setof[2]{\{}{\}}{#1\,|\,#2}
\newcommand{\vx}{\underline{x}}
\renewcommand{mod}[1]{\text{(mod }{#1})}
\newcommand{\Slv}[3]{\sum_{{#2}=1}^{{#3}} {#1}_{{#2}} \v_{{#2}}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 6.2 Riemann-Stieltjes Integrals
Given bounded functions \(f,g\in\mathcal{B}[a,b]\text{,}\) the Riemann-Stieltjes Integral \(\int_a^b f(x)\,dg(x)\text{,}\) when it exists will represent a “weighted sum” of \(f\) over \([a,b]\) with weights determined by “the derivative” of \(g\text{.}\) (although \(g\) won't always be differentiable!) Under suitible conditions, we will have
\begin{equation*}
\int_a^b f(x)\,dg(x)=\int_a^bf(x)g'(x).
\end{equation*}
Roughly, we'll define \(\int_a^b\,dg(X)\) as a limit of “weighted Riemann sums.”
Definition 6.23 .
For functions \(f,g\in\mathcal{B}[a,b]\text{,}\) a partition \(P=\{{a=x_0<\dots<x_n=b}\}\) and an evaluation sequence \(X\) for \(P\text{,}\) define the Riemann-Stieltjes Sum
\begin{equation*}
I(f,P,X,g)=\sum_{i=1}^nf(x_i^*)(g(x_i)-g(x_{i-1})).
\end{equation*}
Definition 6.25 .
For bounded functions \(f,g:[a,b]\rightarrow\mathbb{R}\text{,}\) we say that \(f\) is Riemann-Stieltjes integrable over \([a,b]\) with respect to \(g\) if there is an \(A\in\mathbb{R}\) so that for all \(\varepsilon>0\) there is a partition \(P\in\mathcal{P}[a,b]\) so that for all partitions \(Q\in\mathcal{P}[a,b]\) with \(P\subseteq Q\) and evaluation sequences \(X\) for \(Q\text{,}\)
\begin{equation*}
\left|{I(f,P,X,g)-A}\right|<\varepsilon
\end{equation*}
in which case \(A\) is unique and we call this \(A\) the Riemann-Stieltjes integral of \(f\) over \([a,b]\) with respect to \(g\) and we write
\begin{equation*}
A=\int_a^b f(x)\,dg(x).
\end{equation*}
Let \(\mathcal{R}([a,b],g)\) denote all bounded functions \(f\) which are Riemann-Stieltjes integrable over \([a,b]\) with respect to \(g\text{.}\)
Fact 6.26 .
If \(g(x)=x\text{,}\) then \(\mathcal{R}[a,b]=\mathcal{R}([a,b],g)\text{,}\) and for all \(f\in \mathcal{R}[a,b]\)
\begin{equation*}
\int_a^b f(x)\,dx=\int_a^b f(x)\,dg(x).
\end{equation*}
Example 6.27 .
The Heaviside function is given by
\begin{equation*}
\begin{aligned}
H(x)
=\begin{cases}
0 & x\leq 0\\
1 & x>0.
\end{cases}
\end{aligned}
\end{equation*}
Recall: a function \(f:\mathbb{R}\rightarrow \mathbb{R}\) is right continuous at \(c\in \mathbb{R}\) if \(\lim_{x\rightarrow c^+}f(x)=f(c)\text{.}\)
Proposition 6.28 .
Let \(a,b,c\in\mathbb{R}\) with \(a\leq C<b\) and let \(g(x)=H(x-c)\text{.}\) If \(f:[a,b]\rightarrow \mathbb{R}\) is right continuous at \(c\text{,}\) then
\begin{equation*}
\int_a^b f(x)\,dg(x)=f(c).
\end{equation*}
Proof.
Let \(\varepsilon>0\text{.}\) Fix \(\delta>0\) such that for all \(x\in(c,c+\delta)\) with \(x\in[a,b]\) we have
\begin{equation*}
\left|{f(x)-f(c)}\right|<\varepsilon.
\end{equation*}
Let \(d\in(c,c+\delta)\cap[a,b]\text{,}\) and consider \(P=\{{a,c,d,b}\}\in\mathcal{P}[a,b]\text{.}\) Now let \(Q\in\mathcal{P}[a,b]\) such that \(Q\supseteq P\) and let \(X\) be an evaluation sequence for \(Q\text{.}\) Then write \(Q=\{{a=x_0<\dots<x_n=b}\}\) and \(X=(x_1^*,\dots,x_n^*)\text{.}\) Let \(k=0,\dots,n-1\) with \(x_k=c\text{.}\) Then
\begin{equation*}
\begin{aligned}
g(x_i)-g(x_{i-1})
=\begin{cases}
0&i\neq k+1\\
1&i=k+1
\end{cases}
\end{aligned}
\end{equation*}
for \(i=1,\dots,n\text{.}\) Then \(I(f,Q,X,g)=\sum_{i=1}^n f(x_i^*)(g(x_i)-g(x_{i-1}))=f(x_k^*)\text{.}\) Since \(C\leq x_{k}^*\leq x_{k+1}\leq d<c+\delta\text{,}\) we have \(\left|{f(x_k^*)-f(c)}\right|<\varepsilon\text{.}\) Then \(\left|{I(f,Q,X,g)-f(c)}\right|<\varepsilon\text{.}\)
Theorem 6.29 . Integration by Parts.
Let \(f,g\in\mathcal{B}[a,b]\text{.}\) Then
\begin{equation*}
f\in\mathcal{R}([a,b],g)\iff g\in\mathcal{R}([a,b],f),
\end{equation*}
and in this case
\begin{equation*}
\int_a^b f(x)\,dg(x)=f(b)g(b)-f(a)g(a)-\int_a^b g(x)\,df(x).
\end{equation*}
Theorem 6.31 .
Suppose \(f\in\mathcal{R}[a,b],g:[a,b]\rightarrow\mathbb{R}\) is differentiable, and \(g'\in\mathcal{R}[a,b]\text{.}\) Then \(f\in\mathcal{R}([a,b],g)\) and
\begin{equation*}
\int_a^b f(x)\,dg(x)=\int_a^b f(x)g'(x)\,dx.
\end{equation*}
Example 6.32 .
\(f(x)=x^2, g(x)=\begin{cases}
x^2 & 0\leq x\le 1\\
e^x & 1<x\le 2\\
1 & 2<x\le 3
\end{cases}\) Compute \(\int_0^3 f(x)\,dg(x)\text{.}\)
Solution .
Have \(g\in \mathcal{R}[0,3]\) as is bounded and has only finitely many discontinuities. Also \(f\) is differentiable and \(f'\in \mathcal{R}[0,3]\text{.}\) So \(g\in\mathcal{R}([0,3],f)\) and
\begin{equation*}
\begin{aligned}
\int_0^3 g(x)\,df(x)
&=\int_0^3 g(x)f'(x)\,dx\\
&=\int_0^1 x^2 2x\,dx + \int_1^2 e^x 2x\,dx+\int_2^32x\,dx\\
&=[\frac{1}{2}x^4]_0^1 +[2xe^x]_1^2-\int_1^22e^x\,dx+[x^2]_2^3\\
&=\frac{1}{2}+4e^2-2e-2e^2+2e+9-4
=\frac{11}{2}+2e^2\\
\implies
\int_0^3 f(x)\,dg(x)
&=f(3)g(3)-f(0)g(0)-\int_0^3 g(x)\,df(x)\\
&=9-(\frac{11}{2}+2e^2)
=\frac{7}{2}-2e^2.
\end{aligned}
\end{equation*}
To get a reasonable theory of integration, we'll want to impose more conditions on \(g\) when defining \(\int_a^bf(x)\,dg(x)\text{.}\)
