Definition:Lebesgue Pre-Measure

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): $4.8$