Riemann integral
A generalization of the concept of a Cauchy integral to a certain class of discontinuous functions; introduced by B. Riemann (1853). Let a function $f$ be given on an interval $[a,b]$ and let $a=x_0<x_1<\dots<x_n=b$, 
, 
. The sum
 | (1) |
 |
where 
, is called the Riemann sum corresponding to the given subdivision of 
by the points 
and to the sample of points 
. The number 
is called the limit of the Riemann sums (1) as 
if for any 
a 
can be found such that 
implies the inequality 
. If the Riemann sums have a finite limit 
as 
, then the function 
is called Riemann integrable over 
, where 
. The limit is known as the definite Riemann integral of 
over 
, and is written as
 | (2) |
When 
then, by definition,
 |
and when 
the integral (2) is defined using the equation
 |
A necessary and sufficient condition for the Riemann integrability of 
over 
is the boundedness of 
on this interval and the zero value of the Lebesgue measure of the set of all points of discontinuity of 
contained in 
.
Properties of the Riemann integral.
1) Every Riemann-integrable function 
on 
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 
).
2) The linearity property: For any constants 
and 
, the integrability over 
of both functions 
and 
implies that the function 
is integrable over this interval, and the equation
 |
holds.
3) The integrability over 
of both functions 
and 
implies that their product 
is integrable over this interval.
4) Additivity: The integrability of a function 
over both intervals 
and 
implies that 
is integrable over 
, and
 |
5) If two functions 
and 
are integrable over 
and if 
for every 
in this interval, then
 |
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 |
Riemann integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_integral&oldid=13576