Definition:Lower Darboux Sum
Definition
Let $\closedint a b$ be a closed real interval.
Let $f: \closedint a b \to \R$ be a bounded real function.
Let $P = \set {x_0, x_1, x_2, \ldots, x_n}$ be a finite subdivision of $\closedint a b$.
For all $\nu \in \set {1, 2, \ldots, n}$, let $m_\nu^{\paren f}$ be the infimum of $f$ on the interval $\closedint {x_{\nu - 1} } {x_\nu}$.
Then:
- $\ds \map {L^{\paren f} } P = \sum_{\nu \mathop = 1}^n m_\nu^{\paren f} \paren {x_\nu - x_{\nu - 1} }$
is called the lower Darboux sum of $f$ on $\closedint a b$ belonging (or with respect) to (the subdivision) $P$.
If there is no ambiguity as to what function is under discussion, $m_\nu$ and $\map L P$ are usually used.
Closed Rectangle
Let $R$ be a closed rectangle in $\R^n$.
Let $f : R \to \R$ be a bounded real-valued function.
Let $P$ be a finite subdivision of $R$.
Let $S$ be the set of subrectangles of $P$.
For every:
- $r = \closedint {a_1} {b_1} \times \dotso \times \closedint {a_n} {b_n} \in S$
define:
- $\ds m_r^{\paren f} = \inf_{x \mathop \in r} \map f x$
- $\ds \map v r = \prod_{1 \mathop \le i \mathop \le n} \paren {b_i - a_i}$
Then:
- $\ds \map {L^{\paren f}} P = \sum_{r \mathop \in S} m_r^{\paren f} \map v r$
is called the lower Darboux sum of $f$ on $R$ with respect to $P$.
Also known as
The notation $\map L {f, P}$ or $\map L {P, f}$ can be used in place of $\map {L^{\paren f} } P$.
The lower Darboux sum is also known as:
- the lower Riemann sum (for Georg Friedrich Bernhard Riemann)
- the lower sum.
Also see
- Results about the lower Darboux sum 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
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 13.2$
- 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $2.5$: The Riemann Integral