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

Measure of finite union finite iff measure of all sets in union finite

Formulation 0

Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that

(i)	$E_1, \ldots, E_N \in \mathcal{F}$ are each a D1109: Measurable set in $M$

Then $\mu(\bigcup_{n = 1}^N E_n) < \infty$ if and only if \begin{equation} \mu(E_1), \ldots, \mu(E_N) < \infty \end{equation}

Formulation 1

Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that

(i)	$E_1, \ldots, E_N \in \mathcal{F}$ are each a D1109: Measurable set in $M$

Then \begin{equation} \mu \left( \bigcup_{n = 1}^N E_n \right) < \infty \quad \iff \quad \mu(E_1), \ldots, \mu(E_N) < \infty \end{equation}

Proofs

Proof 0

Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that

(i)	$E_1, \ldots, E_N \in \mathcal{F}$ are each a D1109: Measurable set in $M$

Result R292: Union is smallest upper bound shows that \begin{equation} E_1, \ldots, E_N \subseteq \bigcup_{n = 1}^N E_n \end{equation} Thus, if $\mu(\bigcup_{n \in \mathbb{N}} E_n) < \infty$, then result R975: Isotonicity of unsigned basic measure allows us to deduce \begin{equation} \mu(E_1), \ldots, \mu(E_n) \leq \mu \left( \bigcup_{n = 1}^N E_n \right) < \infty \end{equation} Conversely, if $\mu(E_1), \ldots, \mu(E_N) < \infty$, then result R979: Countable subadditivity of measure gives the upper bound \begin{equation} \mu \left( \bigcup_{n = 1}^N E_n \right) \leq \sum_{n = 1}^N \mu(E_n) < \infty \end{equation} $\square$