Darboux integral

The Darboux integral is a formulation of integration in real analysis defined using upper and lower sums over partitions of an interval. It provides an order-theoretic approach to integration and is equivalent to the Riemann integral.

Definition

Darboux sums

Let be a bounded function. Let

be a partition of the interval .

For each subinterval of , define:

• the infimum:

• the supremum:

The lower Darboux sum of with respect to is

and the upper Darboux sum is

The Darboux integral

Let denote the set of all partitions of .

The lower Darboux integral of on is defined by

and the upper Darboux integral is defined by

If the upper and lower Darboux integrals are equal, then the Darboux integral of on is defined by their common value, that is,

In this case, function is said to be Darboux-integrable.

See also

• Riemann integral
• Partition

Terminology

en	fr	de	zh		ja
Darboux integral	intégrale de Darboux	Darboux-Integral	Darboux 积分	Darboux 積分	ダルブー積分
partition	partition	Partition	分割	分割	分割
Darboux sum	somme de Darboux	Darboux-Summe	Darboux 和	Darboux 和	ダルブー和
upper Darboux sum	somme supérieure de Darboux	obere Darboux-Summe	上 Darboux 和	上 Darboux 和	上ダルブー和
lower Darboux sum	somme inférieure de Darboux	untere Darboux-Summe	下 Darboux 和	下 Darboux 和	下ダルブー和
upper integral	intégrale supérieure	oberes Integral	上 Darboux 积分	上 Darboux 積分	上ダルブー積分
lower integral	intégrale inférieure	unteres Integral	下 Darboux 积分	下 Darboux 積分	下ダルブー積分