My title - Departamento de Matemática da Universidade do Minho

10 STATISTICAL DESCRIPTION OF ORBITS 68
for any (bounded) continuous function ϕ : X → R. The space C 0 (X, R) of bounded continuous real
valued functions on X, equipped with the sup norm, is a separable Banach space. In particular,
it admits a countable set of points {ϕ n } n∈N
which is dense in its unit sphere. Given that, one
defines, for any couple of Borel probability measures µ and ν, a distance
∞∑
∫ ∫
d (µ, ν) = 2 −n ·
∣ ϕ n dµ − ϕ n dν
∣
n=1
It turns out that d is indeed a metric, and that it induces the weak ∗ topology on Prob. The
important fact (somewhere called ”Helly’s theorem”), which follows from the Ascoli-Arzela theorem
together with the above Riesz-Markov representation theorem, is that Prob, equipped with the
weak ∗ topology, is a compact space: any sequence (µ n ) of Borel probability measures admits a
weakly ∗ convergent subsequence µ ni → µ.
Now, we are in position to prove the existence of invariant probability measures for certain well
behaved dynamical systems.
X
X
Krylov-Bogolyubov theorem. A continuous transformation f : X → X of a metrizable
compact space X admits at least one Borel invariant probability measure.
proof. Take any Borel probability measure µ 0 on X, and inductively define a family of probability
measures µ n by µ n+1 = f ∗ µ n . Consider the family of Cesaro means
µ n = 1 n∑
µ k
n + 1
Since the space of Borel probability measures on a compact metrizable space is compact w.r.t.
weak ∗ convergence, there exist a weakly ∗ convergent subsequence µ ni
→ µ. One then easily sees
that
∫
1 ∑n i
∫
(ϕ ◦ f) dµ = lim
(ϕ ◦ f) dµ k
X
i→∞ n i + 1
k=0
X
1 ∑n i
∫
= lim
ϕdµ k+1
i→∞ n i + 1
k=0
X
1 ∑n i
∫
= lim
ϕdµ k + 1 (∫
∫ )
ϕdµ ni+1 − ϕdµ 0 i→∞ n i + 1
k=0
X n i + 1 X
X
∫
= ϕdµ
X
for any bounded continuous observable ϕ, hence that µ is an invariant measure. □
k=0
10.3 Invariant measures and time averages
The relevance of invariant measures when studying the dynamics of continuous transformations is
due to the following crucial observations.
Invariant measures and time averages. Assume that, for a given point x ∈ X, the time
averages
1
n∑
ϕ (x) = lim ϕ ( f k (x) )
n→∞ n + 1
do exist for any bounded continuos observable ϕ. One easily shows that the funcional Cb 0 (X, R) →
R defined by ϕ ↦→ ϕ (x) is linear, bounded and positive definite. There follows from the Riesz-
Markov representation theorem that there exists a unique Borel probability measure µ x on X such
that
∫
ϕ (x) = ϕdµ x
X
k=0