ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4927 on D1158: Measure space
Binary partition additivity of unsigned basic measure
Formulation 0
Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that
(i) $E, F, G \in \mathcal{F}$ are each a D1109: Measurable set in $M$
(ii) $F, G$ is a D5143: Set partition of $X$
Then \begin{equation} \mu(E) = \mu(E \cap F) + \mu(E \cap G) \end{equation}
Proofs
Proof 0
Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that
(i) $E, F, G \in \mathcal{F}$ are each a D1109: Measurable set in $M$
(ii) $F, G$ is a D5143: Set partition of $X$
This result is a particular case of R4926: Finite partition additivity of unsigned basic measure. $\square$