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A generalization of the concept of a [[Cauchy integral|Cauchy integral]] to a certain class of discontinuous functions; introduced by B. Riemann (1853). Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r0819501.png" /> be given on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r0819502.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r0819503.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r0819504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r0819505.png" />. The sum
+
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r0819506.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
A generalization of the concept of a [[Cauchy integral]] to a certain class of discontinuous functions; introduced by B. Riemann (1853). Consider a function $f$ which is given on an interval $[a,b]$. Let $a=x_0<x_1<\dots<x_n=b$ is a partition (subdivision) of the interval $[a,b]$ and $\Delta x_i = x_i-x_{i-1}$, where $i=1,\dots,n$. The sum
 +
\begin{equation}\label{eq:1}
 +
\sigma = f(\xi_1)\Delta x_1+\dots+f(\xi x_i)\Delta x_i +\dots +f(x_n)\Delta x_n,
 +
\end{equation}
 +
where $x_{i-1}\leq\xi_i\leq x_i$, is called the Riemann sum corresponding to the given partition of $[a,b]$ by the points $x_i$ and to the sample of points $\xi_i$. The number $I$ is called the limit of the Riemann sums \eqref{eq:1} as $\max_i \Delta x_i \to 0$ if for any $\varepsilon>0$ a $\delta>0$ can be found such that $\max_i \Delta x_i < \delta$ implies the inequality $|\sigma - I|<\varepsilon$. If the Riemann sums have a finite limit $I$ as $\max_i \Delta x_i \to 0$, then the function $f$ is called Riemann integrable over $[a,b]$, where $a< b$. The limit is known as the definite Riemann integral of $f$ over $[a,b]$, and is written as
 +
\begin{equation}\label{eq:2}
 +
\int\limits_a^bf(x)\,dx.
 +
\end{equation}
 +
When $a=b$ then, by definition,
 +
\begin{equation}
 +
\int\limits_a^af(x)\,dx = 0,
 +
\end{equation}
 +
and when $a>b$ the integral \eqref{eq:2} is defined using the equation
 +
\begin{equation}
 +
\int\limits_a^bf(x)\,dx = -\int\limits_b^af(x)\,dx.
 +
\end{equation}
 +
A necessary and sufficient condition for the Riemann integrability of $f$ over $[a,b]$ is the boundedness of $f$ on this interval and the zero value of the [[Lebesgue measure]] of the set of all points of discontinuity of $f$ contained in $[a,b]$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r0819507.png" /></td> </tr></table>
 
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r0819508.png" />, is called the Riemann sum corresponding to the given subdivision of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r0819509.png" /> by the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195010.png" /> and to the sample of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195011.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195012.png" /> is called the limit of the Riemann sums (1) as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195013.png" /> if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195014.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195015.png" /> can be found such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195016.png" /> implies the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195017.png" />. If the Riemann sums have a finite limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195018.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195019.png" />, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195020.png" /> is called Riemann integrable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195022.png" />. The limit is known as the definite Riemann integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195023.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195024.png" />, and is written as
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
 
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195026.png" /> then, by definition,
+
==Properties of the Riemann integral==
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195027.png" /></td> </tr></table>
 
 
and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195028.png" /> the integral (2) is defined using the equation
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195029.png" /></td> </tr></table>
 
 
A necessary and sufficient condition for the Riemann integrability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195030.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195031.png" /> is the boundedness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195032.png" /> on this interval and the zero value of the [[Lebesgue measure|Lebesgue measure]] of the set of all points of discontinuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195033.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195034.png" />.
 
 
==Properties of the Riemann integral.==
 
 
 
1) Every Riemann-integrable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195035.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195036.png" /> is also bounded on this interval (the converse is not true: The [[Dirichlet-function|Dirichlet function]] is an example of a bounded and non-integrable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195037.png" />).
 
 
2) The linearity property: For any constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195039.png" />, the integrability over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195040.png" /> of both functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195042.png" /> implies that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195043.png" /> is integrable over this interval, and the equation
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195044.png" /></td> </tr></table>
 
 
holds.
 
 
3) The integrability over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195045.png" /> of both functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195047.png" /> implies that their product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195048.png" /> is integrable over this interval.
 
 
4) Additivity: The integrability of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195049.png" /> over both intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195051.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195052.png" /> is integrable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195053.png" />, and
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195054.png" /></td> </tr></table>
 
 
5) If two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195056.png" /> are integrable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195057.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195058.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195059.png" /> in this interval, then
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195060.png" /></td> </tr></table>
 
 
6) The integrability of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195061.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195062.png" /> implies that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195063.png" /> is integrable over this interval, and the estimate
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195064.png" /></td> </tr></table>
 
 
holds.
 
 
7) The mean-value formula: If two real-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195066.png" /> are integrable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195067.png" />, if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195068.png" /> is non-negative or non-positive everywhere on this interval, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195070.png" /> are the least upper and greatest lower bounds of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195071.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195072.png" />, then a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195073.png" /> can be found, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195074.png" />, such that the formula
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195075.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
 
 
holds. If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195076.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195077.png" />, then this interval will contain a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195078.png" /> such that in formula (3),
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195079.png" /></td> </tr></table>
 
 
8) The second mean-value formula (Bonnet's formula): If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195080.png" /> is real-valued and integrable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195081.png" /> and if a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195082.png" /> is real-valued and monotone on this interval, then a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195083.png" /> can be found in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195084.png" /> such that the formula
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195085.png" /></td> </tr></table>
 
 
holds.
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Riemann,  "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe"  H. Weber (ed.) , ''B. Riemann's Gesammelte Mathematische Werke'' , Dover, reprint  (1953)  pp. 227–271  ((Original: Göttinger Akad. Abh. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081950/r08195086.png" /> (1868)))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Il'in,  E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.D. Kudryavtsev,  "A course in mathematical analysis" , '''1–2''' , Moscow  (1988)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)</TD></TR></table>
 
 
 
 
====Comments====
 
  
 +
#  Every Riemann-integrable function $f$ on $[a,b]$ is also bounded on this interval (the converse is not true: The [[Dirichlet-function|Dirichlet function]] is an example of a bounded and non-integrable function on $[a,b]$).
 +
# The linearity property: For any constants $\alpha$ and $\beta$, the integrability over $[a,b]$ of both functions $f$ and $g$ implies that the function $\alpha f + \beta g$ is integrable over this interval, and the equation \begin{equation} \int\limits_a^b[\alpha f(x) + \beta g(x)]\,dx = \alpha\int\limits_a^bf(x)\,dx + \beta\int\limits_a^bg(x)\,dx \end{equation} holds.
 +
# The integrability over $[a,b]$ of both functions $f$ and $g$ implies that their product $fg$ is integrable over this interval.
 +
# Additivity: The integrability of a function $f$ over both intervals $[a,c]$ and $[c,b]$ implies that $f$ is integrable over $[a,b]$, and \begin{equation} \int\limits_a^bf(x)\,dx = \int\limits_a^cf(x)\,dx + \int\limits_c^bf(x)\,dx. \end{equation}
 +
# If two functions $f$ and $g$ are integrable over $[a,b]$ and if $f(x)\geqslant g(x)$ for every $x$ in this interval, then \begin{equation} \int\limits_a^bf(x)\,dx \geqslant \int\limits_a^bg(x)\,dx. \end{equation}
 +
# The integrability of a function $f$ over $[a,b]$ implies that the function $|f|$ is integrable over this interval, and the estimate \begin{equation} \left|\int\limits_a^bf(x)\,dx\right| \leqslant \int\limits_a^b|f(x)|\,dx \end{equation} holds.
 +
# The mean-value formula: If two real-valued functions $f$ and $g$ are integrable over $[a,b]$, if the function $g$ is non-negative or non-positive everywhere on this interval, and if $M$ and $m$ are the least upper and greatest lower bounds of $f$ on $[a,b]$, then a number $\mu$ can be found, $m\leqslant\mu\leqslant M$, such that the formula \begin{equation}\label{eq:3} \int\limits_a^bf(x)g(x)\,dx = \mu\int\limits_a^bg(x)\,dx, \end{equation} holds. If, in addition, $f$ is continuous on $[a,b]$, then this interval will contain a point $\xi$ such that in formula \eqref{eq:3}, \begin{equation} \mu = f(\xi). \end{equation}
 +
# The second mean-value formula (Bonnet's formula): If a function $f$ is real-valued and integrable over $[a,b]$ and if a function $g$ is real-valued and monotone on this interval, then a point $\xi$ can be found in $[a,b]$ such that the formula \begin{equation} \int\limits_a^bf(x)g(x)\,dx = g(a)\int\limits_a^{\xi}f(x)\,dx + g(b)\int\limits_{\xi}^bf(x)\,dx, \end{equation} holds.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.E. Shilov,  "Mathematical analysis" , '''1–2''' , M.I.T.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.N. Pesin,  "Classical and modern integration theories" , Acad. Press  (1970)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1976)  pp. 75–78</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  B. Riemann,  "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe"  H. Weber (ed.) , ''B. Riemann's Gesammelte Mathematische Werke'' , Dover, reprint  (1953)  pp. 227–271  ((Original: Göttinger Akad. Abh. '''13''' (1868)))</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Il'in,  E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1982)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  L.D. Kudryavtsev,  "A course in mathematical analysis" , '''1–2''' , Moscow  (1988)  (In Russian)</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G.E. Shilov,  "Mathematical analysis" , '''1–2''' , M.I.T.  (1974)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  I.N. Pesin,  "Classical and modern integration theories" , Acad. Press  (1970)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1976)  pp. 75–78</TD></TR>
 +
</table>

Latest revision as of 08:25, 25 April 2016


A generalization of the concept of a Cauchy integral to a certain class of discontinuous functions; introduced by B. Riemann (1853). Consider a function $f$ which is given on an interval $[a,b]$. Let $a=x_0<x_1<\dots<x_n=b$ is a partition (subdivision) of the interval $[a,b]$ and $\Delta x_i = x_i-x_{i-1}$, where $i=1,\dots,n$. The sum \begin{equation}\label{eq:1} \sigma = f(\xi_1)\Delta x_1+\dots+f(\xi x_i)\Delta x_i +\dots +f(x_n)\Delta x_n, \end{equation} where $x_{i-1}\leq\xi_i\leq x_i$, is called the Riemann sum corresponding to the given partition of $[a,b]$ by the points $x_i$ and to the sample of points $\xi_i$. The number $I$ is called the limit of the Riemann sums \eqref{eq:1} as $\max_i \Delta x_i \to 0$ if for any $\varepsilon>0$ a $\delta>0$ can be found such that $\max_i \Delta x_i < \delta$ implies the inequality $|\sigma - I|<\varepsilon$. If the Riemann sums have a finite limit $I$ as $\max_i \Delta x_i \to 0$, then the function $f$ is called Riemann integrable over $[a,b]$, where $a< b$. The limit is known as the definite Riemann integral of $f$ over $[a,b]$, and is written as \begin{equation}\label{eq:2} \int\limits_a^bf(x)\,dx. \end{equation} When $a=b$ then, by definition, \begin{equation} \int\limits_a^af(x)\,dx = 0, \end{equation} and when $a>b$ the integral \eqref{eq:2} is defined using the equation \begin{equation} \int\limits_a^bf(x)\,dx = -\int\limits_b^af(x)\,dx. \end{equation} A necessary and sufficient condition for the Riemann integrability of $f$ over $[a,b]$ is the boundedness of $f$ on this interval and the zero value of the Lebesgue measure of the set of all points of discontinuity of $f$ contained in $[a,b]$.



Properties of the Riemann integral

  1. Every Riemann-integrable function $f$ on $[a,b]$ is also bounded on this interval (the converse is not true: The Dirichlet function is an example of a bounded and non-integrable function on $[a,b]$).
  2. The linearity property: For any constants $\alpha$ and $\beta$, the integrability over $[a,b]$ of both functions $f$ and $g$ implies that the function $\alpha f + \beta g$ is integrable over this interval, and the equation \begin{equation} \int\limits_a^b[\alpha f(x) + \beta g(x)]\,dx = \alpha\int\limits_a^bf(x)\,dx + \beta\int\limits_a^bg(x)\,dx \end{equation} holds.
  3. The integrability over $[a,b]$ of both functions $f$ and $g$ implies that their product $fg$ is integrable over this interval.
  4. Additivity: The integrability of a function $f$ over both intervals $[a,c]$ and $[c,b]$ implies that $f$ is integrable over $[a,b]$, and \begin{equation} \int\limits_a^bf(x)\,dx = \int\limits_a^cf(x)\,dx + \int\limits_c^bf(x)\,dx. \end{equation}
  5. If two functions $f$ and $g$ are integrable over $[a,b]$ and if $f(x)\geqslant g(x)$ for every $x$ in this interval, then \begin{equation} \int\limits_a^bf(x)\,dx \geqslant \int\limits_a^bg(x)\,dx. \end{equation}
  6. The integrability of a function $f$ over $[a,b]$ implies that the function $|f|$ is integrable over this interval, and the estimate \begin{equation} \left|\int\limits_a^bf(x)\,dx\right| \leqslant \int\limits_a^b|f(x)|\,dx \end{equation} holds.
  7. The mean-value formula: If two real-valued functions $f$ and $g$ are integrable over $[a,b]$, if the function $g$ is non-negative or non-positive everywhere on this interval, and if $M$ and $m$ are the least upper and greatest lower bounds of $f$ on $[a,b]$, then a number $\mu$ can be found, $m\leqslant\mu\leqslant M$, such that the formula \begin{equation}\label{eq:3} \int\limits_a^bf(x)g(x)\,dx = \mu\int\limits_a^bg(x)\,dx, \end{equation} holds. If, in addition, $f$ is continuous on $[a,b]$, then this interval will contain a point $\xi$ such that in formula \eqref{eq:3}, \begin{equation} \mu = f(\xi). \end{equation}
  8. The second mean-value formula (Bonnet's formula): If a function $f$ is real-valued and integrable over $[a,b]$ and if a function $g$ is real-valued and monotone on this interval, then a point $\xi$ can be found in $[a,b]$ such that the formula \begin{equation} \int\limits_a^bf(x)g(x)\,dx = g(a)\int\limits_a^{\xi}f(x)\,dx + g(b)\int\limits_{\xi}^bf(x)\,dx, \end{equation} holds.

References

[1] B. Riemann, "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" H. Weber (ed.) , B. Riemann's Gesammelte Mathematische Werke , Dover, reprint (1953) pp. 227–271 ((Original: Göttinger Akad. Abh. 13 (1868)))
[2] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)
[3] L.D. Kudryavtsev, "A course in mathematical analysis" , 1–2 , Moscow (1988) (In Russian)
[4] S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)
[a1] G.E. Shilov, "Mathematical analysis" , 1–2 , M.I.T. (1974) (Translated from Russian)
[a2] I.N. Pesin, "Classical and modern integration theories" , Acad. Press (1970) (Translated from Russian)
[a3] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)
[a4] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 75–78
How to Cite This Entry:
Riemann integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_integral&oldid=13576
This article was adapted from an original article by V.A. Il'in (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article