ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4317 on D1158: Measure space
Inclusion-exclusion principle for unsigned basic measure of ternary union
Formulation 0
Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space.
Let $E_1, E_2, E_3 \in \mathcal{F}$ each be a D1109: Measurable set in $M$ such that
(i) \begin{equation} \mu(E_1), \mu(E_2), \mu(E_3) < \infty \end{equation}
Then \begin{equation} \begin{split} \mu(E_1 \cup E_2 \cup E_3) & = \mu(E_1) + \mu(E_2) + \mu(E_3) \\ & \quad - \mu(E_1 \cap E_2) - \mu(E_1 \cap E_3) - \mu(E_2 \cap E_3) \\ & \quad + \mu(E_1 \cap E_2 \cap E_3) \end{split} \end{equation}
Proofs
Proof 0
Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space.
Let $E_1, E_2, E_3 \in \mathcal{F}$ each be a D1109: Measurable set in $M$ such that
(i) \begin{equation} \mu(E_1), \mu(E_2), \mu(E_3) < \infty \end{equation}
This result is a particular case of R2086: Finite inclusion-exclusion principle for unsigned basic measure. $\square$