[211.2.1] In classical mechanics the commutative algebra of observables is the algebra of continuous functions on phase space , that vanish at infinitye (This is a footnote:) e This means that for each and there is a compact subset such that for all .. [211.2.2] The characters (pure states) are point measures on phase space , and one has the isomorphism . [211.2.3] By the Riesz representation theorem the states in classical mechanics are probability measures on phase space . [211.2.4] Every state gives rise to a probability measure space where is the -algebra of measurable subsets of phase space .
[211.3.1] Let be an invertible map such that and are both measurable, i.e. such that where for . [211.3.2] The map is called a measure preserving transformation and the measure on is called invariant under , if for all . [page 212, §0] [212.0.1] An invariant measure is called ergodic with respect to , if it cannot be decomposed into a convex combination of -invariant measures.
[212.1.1] Here and in the following the measure preserving transformation is the adjoint time evolution which is denoted more briefly as . [212.1.2] Pure states (characters) are not invariant under . [212.1.3] Examples of invariant probability measures are furnished by the set of equilibrium states of a conservative system with Hamiltonian dynamics. [212.1.4] If is an equilibrium state of a conservative system then is a measure preserving system.