\begin{definition}[Definition:Lebesgue Pre-Measure]
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.
\end{definition}