ThmDex – An index of mathematical definitions, results, and conjectures.
Countable partition additivity of probability measure
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E, F_0, F_1, F_2, \dots \in \mathcal{F}$ are each an D1716: Event in $P$
(ii) $F_0, F_1, F_2, \ldots$ is a D5143: Set partition of $\Omega$
Then \begin{equation} \mathbb{P}(E) = \sum_{n \in \mathbb{N}} \mathbb{P}(E \cap F_n) \end{equation}
Subresults
R4928: Finite partition additivity of probability measure
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E, F_0, F_1, F_2, \dots \in \mathcal{F}$ are each an D1716: Event in $P$
(ii) $F_0, F_1, F_2, \ldots$ is a D5143: Set partition of $\Omega$
This result is a particular case of R3645: Countable partition additivity of unsigned basic measure. $\square$