# 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.

## Source of Name

This entry was named for Henri Léon Lebesgue.