ThmDex – An index of mathematical definitions, results, and conjectures.
Result R3645 on D1158: Measure space
Countable partition additivity of unsigned basic measure

Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that
 (i) $E, F_0, F_1, F_2, \dots \in \mathcal{F}$ are each a D1109: Measurable set in $M$ (ii) $F_0, F_1, F_2, \ldots$ is a D5143: Set partition of $X$
Then $$\mu(E) = \sum_{n \in \mathbb{N}} \mu(E \cap F_n)$$
Proofs
Proof 0
Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that
 (i) $E, F_0, F_1, F_2, \dots \in \mathcal{F}$ are each a D1109: Measurable set in $M$ (ii) $F_0, F_1, F_2, \ldots$ is a D5143: Set partition of $X$
Since $F_0, F_1, F_2, \dots$ partitions $X$, then applying disjoint additivity and result R237: Intersection distributes over union, one has $$\begin{split} \mu (E) = \mu (E \cap X) & = \mu \left( E \cap \bigcup_{n \in \mathbb{N}} F_n \right) \\ & = \mu \left( \bigcup_{n \in \mathbb{N}} (E \cap F_n) \right) = \sum_{n \in \mathbb{N}} \mu (E \cap F_n) \end{split}$$ $\square$