Definition:Lower Darboux Integral

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Definition

Let $\closedint a b$ be a closed real interval.

Let $f: \closedint a b \to \R$ be a bounded real function.


The lower Darboux integral of $f$ over $\closedint a b$ is defined as:

$\ds \underline {\int_a^b} \map f x \rd x = \sup_P \map L P$

where:

the supremum is taken over all subdivisions $P$ of $\closedint a b$
$\map L P$ denotes the lower Darboux sum of $f$ on $\closedint a b$ belonging to $P$.


Closed Rectangle

Let $R$ be a closed rectangle in $\R^n$.

Let $f : R \to \R$ be a bounded real-valued function on $R$.


The lower integral of $f$ over $R$ is defined as:

$\ds \underline{\int_R} \map f x \rd x = \sup_P \map L P$

where:

$P$ ranges over all finite subdivisions of $R$.
$\map L P$ denotes the lower Darboux sum of $f$ on $R$ with respect to $P$.


Also known as

The lower Darboux integral is also known just as the lower integral.


Also see

  • Results about the lower Darboux integral can be found here.


Source of Name

This entry was named for Jean-Gaston Darboux.


Sources