Definition:Lebesgue Pre-Measure
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Definition
Let $\JJ_{ho}$ be the collection of half-open $n$-rectangles.
$n$-dimensional Lebesgue pre-measure is the mapping $\lambda^n: \JJ_{ho} \to \overline \R_{\ge 0}$ given by:
- $\ds \map {\lambda^n} {\horectr {\mathbf a} {\mathbf b} } = \prod_{i \mathop = 1}^n \paren {b_i - a_i}$
where $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers.
Also see
- Lebesgue Pre-Measure is Pre-Measure
- Lebesgue Measure, the extension of $\lambda^n$ to the Borel $\sigma$-algebra $\map \BB {\R^n}$
Source of Name
This entry was named for Henri Léon Lebesgue.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $4.8$