ThmDex – An index of mathematical definitions, results, and conjectures.
Result R2975 on D1741: Lebesgue sigma-algebra
Lebesgue sigma-algebra is closed under reflections, translations, and scaling
Formulation 0
Let $M = (\mathbb{R}^N, \mathcal{L})$ be a D1742: Lebesgue measurable space such that
(i) $E \in \mathcal{L}$ is a D1779: Lebesgue set in $M$
(ii) $x_0 \in \mathbb{R}^N$ is a D4924: Euclidean real number
Let $a \in \mathbb{R}$ be a D993: Real number.
Then
(1) \begin{equation} - E \in \mathcal{L} \end{equation}
(2) \begin{equation} E + x_0 \in \mathcal{L} \end{equation}
(3) \begin{equation} a E \in \mathcal{L} \end{equation}