Riemann multiple integral
If we say that is Riemann-integrable on and we define the Riemann integral of :
Clearly one has . Also one has when and are any two polyrectangles containing the support of . In fact one can always find a common refinement of both and so that . So, to prove that a function is Riemann-integrable it is enough to prove that for every there exists a polyrectangle such that .
is Riemann measurable on (as defined above). Moreover we define the Peano-Jordan measure of as
When the Peano Jordan measure of is called the volume of , and when the Peano Jordan measure of is called the area of .
Let now be a Riemann measurable set and let be a bounded function. We say that is Riemann measurable if the function
is Riemann integrable as defined before. In this case we denote with
the Riemann integral of on .
|Title||Riemann multiple integral|
|Date of creation||2013-03-22 15:03:34|
|Last modified on||2013-03-22 15:03:34|
|Last modified by||paolini (1187)|