My title - Departamento de Matemática da Universidade do Minho

10 STATISTICAL DESCRIPTION OF ORBITS 67
A particularly important tool, taken from the theory of probability, is the conditional mean.
Let (X, E, µ) be a probability space, and let F be a sub-σ-algebra of E. Given an integrable random
variable ϕ, there exists a unique random variable ϕ F , called the conditional mean of ϕ w.r.t. F,
which is F-measurable (i.e. the inverse image of any Borel set belongs to F) and such that
∫ ∫
ϕ F dµ = ϕdµ
A
for any A ∈ F. Indeed, if ϕ ≥ 0, then one can define ϕ F as equal to the Radon-Nikodym derivative
of the measure A ↦→ ∫ A ϕdµ, defined on F, with respect to the restriction µ | F . The general case is
treated by linearity, writing ϕ as a difference of two non-negative random variables. Uniqueness is
intended µ-a.e., i.e. modulo sets of zero probability. The conditional mean is monotone, namely if
ϕ ≥ 0 then ϕ F ≥ 0, and preserves the mean value, since ∫ X ϕ Fdµ = ∫ ϕdµ. It can be considered
X
as a ”projection” of ϕ onto the space of F-measurable random variable, preserving the mean value.
In particular, if N is the trivial σ-algebra made of events of measure 0 or 1, then ϕ N is constant
a.e. and equal to ∫ X ϕdµ.
A
Topological dynamical systems and Borel measures. If we are interested in the dynamics
of a continuous transformation f : X → X of a topological space X, it is natural to consider
the Borel σ-algebra B, the smallest σ-algebra of subsets of X which contain all open sets. The
map f is then an endomorphism of (X, B). Probability measures on B are said Borel probability
measures. If, moreover, X is a compact metric space, one can consider the space C 0 (X, R) of
bounded continuous real valued functions of X (observe that, since X is compact, any continuous
function is automatically bounded), equipped with the sup norm
‖ϕ − ψ‖ ∞
= sup |ϕ (x) − ψ (x)|
x∈X
These observables are clearly integrable w.r.t. to any Borel probability measure µ, and the mean
value map
∫
ϕ ↦→ ϕdµ
is a bounded, positive definite (in the sense that ∫ ϕdµ ≥ 0 for any ϕ ≥ 0) linear functional on
X
C 0 (X, R). The basic fact about Borel measures is the converse of that, namely
X
Riesz-Markov representation theorem . Let X be a compact metric space. Given any
bounded and positive definite linear functional L on C 0 (X, R) such that L (1) = 1, there exists a
unique Borel probability measures µ such that
∫
L (ϕ) = ϕdµ
for any ϕ ∈ C 0 (X, R).
X
The space of invariant probability measures. The space Prob of probability measures on
a measurable space (X, E) has a natural convex structure: convex combinations of probability
measures are also probability measures. An arbitrary measurable transformation f : X → X of a
measurable space may not admit any invariant probability measure. On the other side, if µ 0 and
µ 1 are invariant probability measures, so are their convex combinations µ t = (1 − t) µ 0 + tµ 1 for
any t ∈ [0, 1]. This means that the set Prob f of invariant probabilty measures on E is a convex
set: if it contains two points, it contains the whole segment between them.
Now, let (X, d) be a compact metric space and let B its Borel σ-algebra. The space Prob of
probability measures on B can be equipped with a natural topology, called the weak ∗ topology,
which says essentially that two measures are near if they give nearby mean values to some well
behaved observables. Formally, one says that a sequence of measures (µ n ) converge weakly* to a
measure µ, which we denote simply as µ n → µ, if
∫ ∫
ϕdµ n → ϕdµ
X
X