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 $\JJ_{ho}^n$.


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


Proof

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

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

Also, from Characterization of Euclidean Borel Sigma-Algebra, $\map \BB {\R^n} = \map \sigma {\JJ_{ho}^n}$.


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

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

$\map {\lambda^n} {\horectr {-n} n} = \ds \prod_{i \mathop = 1}^n n - \paren {-n} = \paren {2 n}^n < +\infty$


Hence, Carathéodory's Theorem and its corollary apply.

These yield existence and uniqueness of Lebesgue measure, the extension of $\lambda^n$ to $\map \BB {\R^n}$.

$\blacksquare$


Sources

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