Editing Quantum logic (section)

=== Mackey observables ===
Given a [[Orthocomplement|orthocomplemented lattice]] ''Q'', a Mackey observable φ is a [[Countably additive measure|countably additive homomorphism]] from the orthocomplemented lattice of Borel subsets of '''R''' to ''Q''.  In symbols, this means that for any sequence {''S''<sub>''i''</sub>}<sub>''i''</sub> of pairwise-disjoint Borel subsets of '''R''', {φ(''S''<sub>''i''</sub>)}<sub>''i''</sub> are pairwise-orthogonal propositions (elements of ''Q'') and
: <math> \varphi\left(\bigcup_{i=1}^\infty S_i\right) = \sum_{i=1}^\infty \varphi(S_i). </math>

Equivalently, a Mackey observable is a [[projection-valued measure]] on '''R'''.

'''Theorem''' ([[Spectral theorem]]). If ''Q'' is the lattice of closed subspaces of Hilbert ''H'', then there is a bijective correspondence between Mackey observables and densely-defined self-adjoint operators on ''H''.