Existence and Uniqueness of Lebesgue Measure

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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.


Proof

From Lebesgue Pre-Measure is Pre-Measure, $\lambda^n$ is a pre-measure on $\mathcal{J}_{ho}^n$.

By Half-Open Rectangles form Semiring of Sets, $\mathcal{J}_{ho}^n$ is a semiring of sets.

Also, from Characterization of Euclidean Borel Sigma-Algebra, $\mathcal B \left({\R^n}\right) = \sigma \left({\mathcal{J}_{ho}^n}\right)$.


Now observe that the half-open $n$-rectangles $\left[[{-n, n}\right))$ form an increasing sequence of sets with limit $\R^n$.

Also, by definition of $\lambda^n$, have:

$\lambda^n \left({\left[[{-n, n}\right))}\right) = \displaystyle \prod_{i \mathop = 1}^n n - \left({-n}\right) = \left({2n}\right)^n < +\infty$


Hence, Carathéodory's Theorem and its corollary apply.

These yield existence and uniqueness of Lebesgue measure, the extension of $\lambda^n$ to $\mathcal B \left({\R^n}\right)$.

$\blacksquare$


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