ThmDex – An index of mathematical definitions, results, and conjectures.
Result R1164 on D1744: Lebesgue measure space
Translation invariance of Lebesgue measure
Formulation 0
Let $M = (\mathbb{R}^n, \mathcal{L}, \mu)$ be a D1744: Lebesgue measure space.
Let $x \in \mathbb{R}^n$ be a D4924: Euclidean real number.
Then
(1) $\forall \, E \in \mathcal{L} : E + x \in \mathcal{L}$
(2) $\forall \, E \in \mathcal{L} : \mu(E + x) = \mu(E)$
Proofs
Proof 0
Let $M = (\mathbb{R}^n, \mathcal{L}, \mu)$ be a D1744: Lebesgue measure space.
Let $x \in \mathbb{R}^n$ be a D4924: Euclidean real number.