Darboux integral
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In real analysis, a =
branch of=20
mathematics, the =
Darboux=20
integral or Darboux sum is one possible definition of the
[edit]=20 Definition
A partition = of an=20 interval [a,b] is a finite sequence of values=20 xi such that
=20
Each interval = [xi=E2=88=921,xi] is=20 called a subinterval of the partition. Let = =C6=92:[a,b]=E2=86=92R=20 be a bounded function, and let
=20
be a partition of [a,b]. Let
=20
![3D""=20](3D"http://en.wikipedia.org/wiki/File:Darboux.svg")
=20
The upper Darboux sum of =C6=92 with respect to P = is
=20
The lower Darboux sum of =C6=92 with respect to P = is
=20
The upper Darboux integral of =C6=92 is
=20
The lower Darboux integral of =C6=92 is
=20
If U=C6=92 =3D L=C6=92, = then we say that =C6=92=20 is Darboux-integrable and set
=20
the common value of the upper and lower Darboux integrals.
[edit] =20
Facts =
about the=20
Darboux integral
![3D""](3D"http://en.wikipedia.org/wiki/File:Darboux_refinement.svg")
A refinement of the partition
=20
is a partition
=20
such that for every i with
=20
there is an integer r(i) such that
=20
In other words, to make a refinement, cut the subintervals into = smaller=20 pieces and do not remove any existing cuts. If
=20
is a refinement of
=20
then
=20
and
=20
If P1, P2 are two partitions of = the same=20 interval (one need not be a refinement of the other), then
.=20
It follows that
=20
Riemann sums always lie between the corresponding lower and upper = Darboux=20 sums. Formally, if
and
=20
together make a tagged partition
=20
(as in the definition of the Riemann = integral), and=20 if the Riemann sum of =C6=92 corresponding to P and T is = R,=20 then
=20
From the previous fact, Riemann integrals are at least as strong as = Darboux=20 integrals: If the Darboux integral exists, then the upper and lower = Darboux sums=20 corresponding to a sufficiently fine partition will be close to the = value of the=20 integral, so any Riemann sum over the same partition will also be close = to the=20 value of the integral. It is not hard to see that there is a tagged = partition=20 that comes arbitrarily close to the value of the upper Darboux integral = or lower=20 Darboux integral, and consequently, if the Riemann integral exists, then = the=20 Darboux integral must exist as well.
[edit]=20 See also
|
| <=
/TR>
|
|---|---|
| Methods |
Riemann=20
integral =C2=B7 |
| Improper Integrals |
|
| Stochastic integrals |
|
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