ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4926 on D1158: Measure space
Finite partition additivity of unsigned basic measure
Formulation 0
Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that
(i) $E, F_1, \ldots, F_N \in \mathcal{F}$ are each a D1109: Measurable set in $M$
(ii) $F_1, \ldots, F_N$ is a D5143: Set partition of $X$
Then \begin{equation} \mu(E) = \sum_{n = 1}^N \mu(E \cap F_n) \end{equation}
Proofs
Proof 0
Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that
(i) $E, F_1, \ldots, F_N \in \mathcal{F}$ are each a D1109: Measurable set in $M$
(ii) $F_1, \ldots, F_N$ is a D5143: Set partition of $X$
This result is a particular case of R3645: Countable partition additivity of unsigned basic measure. $\square$