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 $\text{[math]}$ be a bounded function. Let

$\text{[math]}$

be a partition of the interval $\text{[math]}$ .

For each subinterval $\text{[math]}$ of $\text{[math]}$ , define:

the infimum:

$\text{[math]}$

the supremum:

$\text{[math]}$

The lower Darboux sum of $\text{[math]}$ with respect to $\text{[math]}$ is

$\text{[math]}$

and the upper Darboux sum is

$\text{[math]}$

The Darboux integral

Let $\text{[math]}$ denote the set of all partitions of $\text{[math]}$ .

The lower Darboux integral of $\text{[math]}$ on $\text{[math]}$ is defined by

$\text{[math]}$

and the upper Darboux integral is defined by

$\text{[math]}$

If the upper and lower Darboux integrals are equal, then the Darboux integral of $\text{[math]}$ on $\text{[math]}$ is defined by their common value, that is,

$\text{[math]}$

In this case, function $\text{[math]}$ is said to be Darboux-integrable.

Definition

Darboux sums

The Darboux integral

See also