Existence and Uniqueness of Lebesgue Measure

Theorem
Let $\lambda^n$ be the Lebesgue pre-measure on the half-open $n$-rectangles $\mathcal{J}_{ho}^n$.

Then Lebesgue measure, the extension of $\lambda^n$ to the Borel $\sigma$-algebra $\mathcal B \left({\R^n}\right)$, exists and is unique.