ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4277 on D1158: Measure space
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 $F : = X$. $\square$