# 5 Conservative Systems

[211.2.1] In classical mechanics the commutative algebra of
observables A=C0Γ is the
algebra of continuous functions on
phase space Γ, that vanish at infinity.
[211.2.2] The characters (pure states) are point measures
on phase space Γ, and one has the
isomorphism Γ≡XA.
[211.2.3] By the Riesz representation theorem the states
μ∈A*≡C0XA*≡C0Γ*
in classical mechanics are probability measures on phase space
Γ≡XA.
[211.2.4] Every state μ∈C0Γ* gives
rise to a probability measure space
Γ,G,μ
where G is the σ-algebra of measurable subsets of
phase space Γ.

[211.3.1] Let S:Γ→Γ be an invertible map
such that S and S-1 are both measurable,
i.e. such that S-1G=SG=G
where
SG:=Sx:x∈G for G∈G.
[211.3.2] The map S is called a measure preserving transformation
and the measure μ on Γ is called
invariant under S, if
μG=μSG=μS-1G
for all G∈G.
[page 212, §0]
[212.0.1] An invariant measure is called ergodic
with respect to S, if it cannot
be decomposed into a convex combination of S-invariant measures.

[212.1.1] Here and in the following
the measure preserving transformation is
the adjoint time evolution A*Tt˙
which is denoted more briefly as ΓTt˙=A*Tt˙.
[212.1.2] Pure states (characters) are not invariant under ΓTt˙.
[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 μ is an equilibrium state of a conservative system
then Γ,G,μ,ΓTt˙ is a measure preserving
system.