Definition:Lebesgue Pre-Measure

Definition
Let $\mathcal J_{ho}$ be the collection of half-open $n$-rectangles.

$n$-dimensional Lebesgue pre-measure is the mapping $\lambda^n: \mathcal J_{ho} \to \overline \R_{\ge 0}$ given by:


 * $\displaystyle \lambda^n \left({ \left[[{\mathbf a \,.\,.\, \mathbf b}\right)) }\right) = \prod_{i \mathop = 1}^n \left({b_i - a_i}\right)$

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 $\mathcal B \left({\R^n}\right)$