ThmDex – An index of mathematical definitions, results, and conjectures.
Result R1166 on D1744: Lebesgue measure space
Lebesgue measure under scaling
Formulation 0
Let $M = (\mathbb{R}^N, \mathcal{L}, \ell)$ be a D1744: Lebesgue measure space such that
(i) $E \in \mathcal{L}$ is a D1779: Lebesgue set in $M$
Let $a \in \mathbb{R} \setminus \{ 0 \}$ be a D993: Real number.
Then
(1) \begin{equation} a E \in \mathcal{L} \end{equation}
(2) \begin{equation} \ell(a E) = |a|^N \ell(E) \end{equation}
Proofs
Proof 0
Let $M = (\mathbb{R}^N, \mathcal{L}, \ell)$ be a D1744: Lebesgue measure space such that
(i) $E \in \mathcal{L}$ is a D1779: Lebesgue set in $M$
Let $a \in \mathbb{R} \setminus \{ 0 \}$ be a D993: Real number.