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
- 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous) ... (next): $6.1$
- 1970: Arne Broman: Introduction to Partial Differential Equations ... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.3$ Definitions
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Darboux's theorem
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Darboux's theorem
- 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $2.5$: The Riemann Integral