\begin{definition}[Definition:Lebesgue Measure]
Let $\JJ_{ho}^n$ be the set of half-open $n$-rectangles.
Let $\map \BB {\R^n}$ be the Borel $\sigma$-algebra on $\R^n$.
Let $\lambda^n$ be the $n$-dimensional Lebesgue pre-measure on $\JJ_{ho}^n$.
Any measure $\mu$ extending $\lambda^n$ to $\map \BB {\R^n}$ is called '''$n$-dimensional Lebesgue measure'''.
That is, $\mu$ is an '''$n$-dimensional Lebesgue measure''' {{iff}} it satisfies:
:$\mu \restriction_{\JJ_{ho}^n} = \lambda^n$
where $\restriction$ denotes restriction.
Usually, this measure is also denoted by $\lambda^n$, even though this may be considered abuse of notation.
{{refactor|Below characterisation of Lebesgue measure $\lambda^1$ is reminiscent of the proof of Carathéodory's Theorem (Measure Theory) as discussed on User:Lord_Farin/Sandbox/Archive, and I wonder if it adds anything}}
\end{definition}