Upper and lower Darboux integrals

Definition

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be bounded. The upper Darboux integral of $f$ is the infimum of the upper Darboux sums $U(f, P)$ over all partitions $P$ of $[a, b]$ . The lower Darboux integral of $f$ is the supremum of the lower Darboux sums $L(f, P)$ over all partitions $P$ .

Remarks

The lower Darboux integral is at most the upper one. The function is Darboux integrable if the two are equal. By a theorem, this is the same as Riemann integrability. For the Dirichlet function on $[0, 1]$ , the upper integral is $1$ and the lower integral is $0$ .

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Partition of an interval definition
Sequence definition
Real sequence definition
Bounded sequence definition
Upper and lower Darboux sums definition
Upper and lower Darboux integrals definition

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Tag	`09FQT057T7`
URI	`m2://09FQT057T7`
Kind	definition
Writing	W0 stub
MSC	26A42
Defines	Upper and lower Darboux integrals
Uses	Upper and lower Darboux sums

Upper and lower Darboux integrals

Definition

Remarks

Reading path

Where this leads

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