My title - Departamento de Matemática da Universidade do Minho

10 STATISTICAL DESCRIPTION OF ORBITS 65
Lebesgue density theorem.
the density
Let A ⊂ R n be a Lebesgue-measurable set. For l-almos any x ∈ A
l(A ∩ B ε (x))
lim
= 1
ε→0 l(B ε (x))
Kolmogorov extension. Let X be a finite space, equipped with the discrete topology, and
let Σ + be the topological product X N = {x : N → X}, its point indentified with sequences x =
(x 1 , x 2 ..., x n , ...) with x n ∈ X. Let C be the collection of cylinders of X, the subsets of the form
C B = { x ∈ Σ + s.t. (x 1 , x 2 ..., x n ) ∈ B }
with B an open subset of X n . Cylinders form a basis of the product topology of Σ + , which
makes Σ + a compact metrizable space. In particular, the Borel σ-álgebra of Σ + is B = σ (C).
Let µ 1 , µ 2 , µ 3 , . . . , µ n , . . . be probability measures defined on the Borel sets of X, X 2 , . . . , X n , . . . ,
respectively. The sequence (µ n ) is said consistent if
µ n+1 (B × X) = µ n (B)
for any n and any Borel subset B ⊂ X n . The (most elementary version of) Kolmogorov extension
theorem says that
Kolmogorov extension theorem. Given a consistent family of probability measures as above,
there exists a unique probability measure µ, defined on the Borel σ-algebra of Σ + , such that
for any cilinder C B .
µ (C B ) = µ n (B)
The proof consists in the following two steps. First, observe that cylinders form an algebra, and
use consistency of the µ n ’s to verify that the formula above does define a function µ : C → [0, 1] on
cylinders (i.e. it does not depend on the different ways the same cylinder may be presented) which
is additive and properly normalized. Then, use compactness of X to check that µ is continuous at
∅, in order to apply Carathéodory theorem. Indeed, let (A n ) be a sequence of cilinders such that
A n ↓ ∅, and assume by contradiction that µ (A n ) > δ > 0 for any n. This implies that A n ≠ ∅ for
any n, but, since the A n are compact, then the Cantor intersection theorem says that ∩ n A n ≠ ∅,
contrary to the hypothesis.
Kolmogorov theorem is the key tool in probability theory, since it allows one to construct measures
which describe an infinite sequence of trials starting with some rule which gives information
about the n-th trial given the knowledge of the first n−1. It actually works with much more general
spaces and in a more general setting. Also, one can easily adapt the construction to ∏ n∈N X n, the
topological product of a countable family of finite spaces. In some precise sense, this is a universal
model of a dynamical system.
If X = {0, 1}, then Σ + = X N is the state space of infinite Bernoulli trials with two possible
outcomes: sucess and failure. Let µ 1 : P (X) → [0, 1] be a any probability measure, defined by
µ 1 ({1}) = p. Kolmogorov construction can be applied postulating the independence of different
trials, i.e. declaring that the family formed by the cilinders {x n = 1} is an independent family,
and giving measure p to each {x n = 1}. The resulting probability space (Σ + , B, µ) describes the
infinite independent Bernoulli trials. Of course, the very same construction can be made when X
is a finite space with any finite numer z of elements.
10.2 Transformations and invariant measures
Measurable transformations. A transformation f : X → X of the measurable space (X, E)
is said measurable if f −1 (A) ∈ E for any A ∈ E. A measurable transformation f is said an
endomorphism of the measurable space, or an automorphism if it is invertible and its inverse is
measurable too.