Darboux Integral

The Darboux integral, also called a Darboux-Stieltjes integral, is a variant of the Stieltjes integral that is defined as a common value for the lower and upper Darboux integrals.

Let 👁 f
and 👁 alpha
be bounded real functions on an interval 👁 [a,b]
, with 👁 alpha
nondecreasing. For any partition 👁 P
given by 👁 a=x_0<x_1<...<x_n=b
, let 👁 delta_r=[x_(r-1),x_r]
.

The lower Darboux integral is the supremum of all lower sums of the form

where 👁 m(f,I)
denotes the infimum of 👁 f(x)
over the interval 👁 I
.

Likewise the upper Darboux integral is the infimum of all upper sums of the form

where 👁 M(f,I)
denotes the supremum of 👁 f(x)
over the interval 👁 I
.

The lower Darboux integral is less or equal to the upper Darboux integral, and that the Darboux integral is a linear form on the vector space of Darboux-integrable functions on 👁 [a,b]
for a given 👁 alpha
.

If 👁 alpha(x)=x
, the original upper and lower Darboux integrals proposed by Darboux in 1875 are recovered.

If the Stieltjes integral exists, then the Darboux integral also exists and has the same value. If 👁 alpha
is continuous, then the two integrals are identical. The Lebesgue integral is a significant extension of the Darboux integral.

The following example shows a difference between the Stieltjes and Darboux integrals. Let 👁 [a,b]=[1,3]
, 👁 f(x)=8
for 👁 1<=x<2
, 👁 f(x)=4
for 👁 2<=x<=3
, 👁 alpha(x)=0
for 👁 1<=x<=2
and 👁 alpha(x)=1
for 👁 2<x<=3
. If 2 belongs to the used partition 👁 P
, then 👁 L(P)=U(P)=4
, and all Riemann sums are 4. If 2 doesn't belong to the partition, then 👁 L(P)=4,U(P)=8
, and the Riemann sums are 4 or 8. Hence the Darboux integral 👁 intf(x)dalpha(x)=4
, but the Riemann integral (defined as the limit of Riemann sums for the mesh size going to zero) doesn't exist.

See also

Lower Integral, Lower Sum, Riemann Integral, Upper Integral, Upper Sum

This entry contributed by Allan Cortzen

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References

Kestelman, H. Modern Theories of Integration, 2nd rev. ed. New York: Dover, p. 250, 1960.

Referenced on Wolfram|Alpha

Darboux Integral

Cite this as:

Cortzen, Allan. "Darboux Integral." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/DarbouxIntegral.html

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