  We apply the algebraic theory of infinite classical lattices from Part I to
write an axiomatic theory of measurements, based on Mackey's axioms for quantum
mechanics. The axioms give a complete theory of measurements in the sense of
Haag and Kastler, taking the traditional form of a logic of propositions
provided with a classical spectral theorem. The results are expressed in terms
of probability distributions of individual measurements. As applications, we
give a separation theorem for states by the set of observables and discuss its
relationship to the equivalence of ensembles in the thermodynamic-limit
program. We also introduce a weak equivalence of states based on the theory.
