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 $\map \BB {\R^n}$