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m (Properties of the Riemann integral.)
m (Properties of the Riemann integral.: some tex)
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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|Lebesgue measure]] of the set of all points of discontinuity of $f$ contained in $[a,b]$.
 
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|Lebesgue measure]] of the set of all points of discontinuity of $f$ contained in $[a,b]$.
  
==Properties of the Riemann integral.==
 
  
  
#  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
 
  
<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>
+
==Properties of the Riemann integral.==
  
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.
+
#  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.  
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
+
# 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}
<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>
+
# 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 <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
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>
 
<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>

Revision as of 06:52, 25 December 2012

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 over implies that the function is integrable over this interval, and the estimate

holds.

7) The mean-value formula: If two real-valued functions and are integrable over , if the function is non-negative or non-positive everywhere on this interval, and if and are the least upper and greatest lower bounds of on , then a number can be found, , such that the formula

(3)

holds. If, in addition, is continuous on , then this interval will contain a point such that in formula (3),

8) The second mean-value formula (Bonnet's formula): If a function is real-valued and integrable over and if a function is real-valued and monotone on this interval, then a point can be found in such that the formula

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. (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)


Comments

References

[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://www.encyclopediaofmath.org/index.php?title=Riemann_integral&oldid=29270
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