ThmDex – An index of mathematical definitions, results, and conjectures.
Isotonicity of Lebesgue measure
Formulation 0
Let $M = (\mathbb{R}^D, \mathcal{L}, \ell)$ be a D1744: Lebesgue measure space such that
(i) $E, F \in \mathcal{L}$ are each a D1779: Lebesgue set in $M$
(ii) \begin{equation} E \subseteq F \end{equation}
Then \begin{equation} \ell(E) \leq \ell(F) \end{equation}
Formulation 1
Let $M = (\mathbb{R}^D, \mathcal{L}, \ell)$ be a D1744: Lebesgue measure space.
Then \begin{equation} \forall \, E, F \in \mathcal{L} \left( E \subseteq F \quad \implies \quad \ell(F) \leq \ell(F) \right) \end{equation}
Proofs
Proof 0
Let $M = (\mathbb{R}^D, \mathcal{L}, \ell)$ be a D1744: Lebesgue measure space such that
(i) $E, F \in \mathcal{L}$ are each a D1779: Lebesgue set in $M$
(ii) \begin{equation} E \subseteq F \end{equation}
This result is a special case of R2168: Isotonicity of Lebesgue outer measure. $\square$