ThmDex – An index of mathematical definitions, results, and conjectures.

Reflection invariance of Lebesgue measure

Formulation 0

Let $M = (\mathbb{R}^n, \mathcal{L}, \mu)$ be a D1744: Lebesgue measure space.

Then

(1)	$\forall \, E \in \mathcal{L} : - E \in \mathcal{L}$
(2)	$\forall \, E \in \mathcal{L} : \mu(-E) = \mu(E)$

Proofs

Proof 0

Let $M = (\mathbb{R}^n, \mathcal{L}, \mu)$ be a D1744: Lebesgue measure space.

This result is a consequence of the results

(i)	R2975: Lebesgue sigma-algebra is closed under reflections, translations, and scaling
(ii)	R2441: Reflection invariance of Lebesgue outer measure

$\square$