My title - Departamento de Matemática da Universidade do Minho

10 STATISTICAL DESCRIPTION OF ORBITS 66
Observe that an endomorphism f of a measurable space (X, E) acts naturally on the space of
measures on E by ”push forward”: if µ is a measure, then f ∗ µ, defined by (f ∗ µ) (A) = µ ( f −1 (A) )
for any A ∈ E, is also a measure.
Let f be an endomorphism of the measurable space (X, E). A probability measure µ on E is
invariant (w.r.t. the transformation f) if f ∗ µ = µ, namely if
µ ( f −1 (A) ) = µ (A)
for any A ∈ E. If this happens, we also say that f is an endomrphism (resp. an automorphism) of
the probability space (X, E, µ). The meaning of this definition is that ”mean values” of integrable
observables ϕ : X → R with respect to invariant probability measures do not change with time, in
the sense that ∫ X ϕdµ = ∫ (ϕ ◦ f) dµ.
X
Given an endomorphism f of the probability space (X, E, µ), one says that an event A ∈ E is
invariant mod 0 if µ ( A △ f −1 (A) ) = 0. The set of invariant mod 0 events form a sub-σ-algebra
of E, denoted by E f .
How to prove that a measure is invariant. The very definition of invariance does not help
too much if we want to prove that a certain measure µ on the σ-algebra E is invariant w.r.t. the
measurable transformation f : X → X. The trick is the following. Suppose that we can prove that
µ ( f −1 (C) ) = µ (C) for any C ∈ C, where C is some subset of E. Caratheodory theorem implies
that f ∗ µ and µ are the same measure on the σ-algebra σ (C) generated by C. On the other side,
the family of those A ∈ E such that µ ( f −1 (A) ) = µ (A) is easily seen to be a σ-algebra. Hence,
if it happens that σ (C) = E, then µ is actually invariant. In other words, in order to prove that
µ is invariant it is sufficient to check that µ ( f −1 (C) ) = µ (C) for any C belonging to a family of
subsets of X which generate the σ-algebra E.
Observables as random variables. When dealing with a endomorphism f : X → X of the
probability space (X, E, µ), one should consider measurable observables ϕ : X → R (or C), those
functions such that ϕ −1 (A) ∈ E for any Borel set A ⊂ R. In the context of probability theory
they are called ”random variables”, and the sequence of observables ϕ ◦ f n may be interpreted as
a ”random process”. If ϕ is integrable, the Lebesgue integral ∫ ϕdµ is interpreted as the ”mean
X
value” of ϕ. Of course, invariance of a measurable observable must be intended modulo sets of zero
measure. Then, one can consider the Banach spaces L p (µ) of (equivalence classes of ) observables
equipped with the L p -norm
(∫ ) 1/p
‖ϕ‖ p
= |ϕ| p dµ
and use the full power of integration theory to get informations about the dynamical system. In
particular, L 2 (µ) is a Hilbert space if equipped with the inner product
∫
〈ϕ, ψ〉 = ϕψdµ
X
Conditional mean. Recall that, given a measurable space (X, E), a measure ν is said absolutely
continuous w.r.t. the measure µ if ν (A) = 0 whenever µ (A) = 0. The following technical result
(which may be proved using Hilbert space techniques) is particularly useful:
Radon-Nikodym theorem. Let (X, E, µ) be a probability space, and let ν be a finite measure
over E which is absolutely continuous with respect to µ. Then there exists a nonnegative integrable
random variable ρ (called the Radon Nikodym derivative of ν w.r.t. µ and denoted by dν/dµ) such
that
∫
ν (A) = ρdµ
for any A ∈ E.
A