Riemann multiple integral

We are going to extend the conceptMathworldPlanetmath of Riemann integral to functions of several variables.

Let f:n be a bounded function with compact support. Recalling the definitions of polyrectangle and the definitions of upper and lower Riemann sums on polyrectangles, we define

S*(f):=inf{S*(f,P):P is a polyrectangle, f(x)=0 for every xnP},
S*(f):=sup{S*(f,P):P is a polyrectangle, f(x)=0 for every xnP}.

If S*(f)=S*(f) we say that f is Riemann-integrable on n and we define the Riemann integral of f:


Clearly one has S*(f,P)S*(f,P). Also one has S*(f,P)S*(f,P) when P and P are any two polyrectangles containing the supportMathworldPlanetmath of f. In fact one can always find a common refinement P′′ of both P and P so that S*(f,P)S*(f,P′′)S*(f,P′′)S*(f,P). So, to prove that a function is Riemann-integrable it is enough to prove that for every ϵ>0 there exists a polyrectangle P such that S*(f,P)-S*(f,P)<ϵ.

Next we are going to define the integral on more general domains. As a byproduct we also define the measureMathworldPlanetmath of sets in n.

Let Dn be a bounded set. We say that D is Riemann measurable if the characteristic functionMathworldPlanetmathPlanetmathPlanetmath

χD(x):={1if xD0otherwise

is Riemann measurable on n (as defined above). Moreover we define the Peano-Jordan measure of D as


When n=3 the Peano Jordan measure of D is called the volume of D, and when n=2 the Peano Jordan measure of D is called the area of D.

Let now Dn be a Riemann measurable setMathworldPlanetmath and let f:D be a bounded function. We say that f is Riemann measurable if the function f¯:n

f¯(x):={f(x)if xD0otherwise

is Riemann integrable as defined before. In this case we denote with


the Riemann integral of f on D.

Title Riemann multiple integral
Canonical name RiemannMultipleIntegral
Date of creation 2013-03-22 15:03:34
Last modified on 2013-03-22 15:03:34
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 14
Author paolini (1187)
Entry type Definition
Classification msc 26A42
Related topic Polyrectangle
Related topic RiemannIntegral
Related topic Integral2
Related topic AreaOfPlaneRegion
Related topic DevelopableSurface
Related topic VolumeAsIntegral
Related topic AreaOfPolygon
Related topic MoscowMathematicalPapyrus
Related topic IntegralOverPlaneRegion
Defines Riemann integrable
Defines Peano Jordan
Defines measurable
Defines area
Defines volume
Defines Jordan content