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

Result R4277 on D1158: Measure space
Subresult of R978: Measure of set difference

Measure of measurable set complement

Formulation 0

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

(i)	$E \in \mathcal{F}$ is a D1109: Measurable set in $M$
(ii)	\begin{equation} \mu(E) < \infty \end{equation}

Then \begin{equation} \mu(X \setminus E) = \mu(X) - \mu(E) \end{equation}

Formulation 1

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

(i)	$E \in \mathcal{F}$ is a D1109: Measurable set in $M$
(ii)	\begin{equation} \mu(E) < \infty \end{equation}

Then \begin{equation} \mu(E^{\complement}) = \mu(X) - \mu(E) \end{equation}

Proofs

Proof 0

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

(i)	$E \in \mathcal{F}$ is a D1109: Measurable set in $M$
(ii)	\begin{equation} \mu(E) < \infty \end{equation}

This result is a particular case of R978: Measure of set difference where $F : = X$. $\square$