My title - Departamento de Matemática da Universidade do Minho

10 STATISTICAL DESCRIPTION OF ORBITS 63
10 Statistical description of orbits
Together with the topological point of view, a source of informations about dynamical systems is
their statistical description. The idea is to measure the relative size of those points whose orbits
have certain definite properties. This is done looking for invariant probability measures, and the
main result is the Birkhoff-Khinchin ergodic theorem. To state and prove the Birkhoff-Khinchin
ergodic theorem, we need to recall many standard facts and results of integration theory. You can
find most of them in the classical manuals by W. Rudin, Real and complex analysis, McGraw-Hill,
New York 1966, or by P. Halmos, Measure theory, Springer-Verlag. New York 1974.
10.1 Probability measures
Probability spaces. A measurable space is a pair (X, E), a non-empty set X together with a σ-
algebra of subsets E. Recall that a (Boolean) algebra is a nonempty family A of subsets of X which
contains X, which contains the complement of any of its elements, and which is closed under finite
unions and intersections. A σ-algebra is an algebra which is also closed under countable unions
and intersections. Given any family C of subsets of X, there exists a minimal σ-algebra σ (C) which
contains all the elements of C, which is called the σ-algebra generated by C.
If (X, τ) is a topological space, the Borel σ-algebra is σ (τ), the smallest σ-algebra which
contains all open sets.
A measure on the measurable space (X, E) is a σ-additive function µ : E → [0, ∞] such that
µ (∅) = 0. Here σ-additivity means that, if (S n ) is a countable family of pairwise disjoint elements
of E, then
µ (∪ n S n ) = ∑ µ (S n )
n
The triple (Ω, E, µ) is said a measure space, or probability space if it happens that µ (X) = 1. Given
a probability space, measurable sets A ∈ E are commonly called ”events”, and the number µ (A)
is called ”probability of the event A”. Basic properties of probability measures are the following:
probability measures are monotone, i.e. µ(S) ≤ µ(T ) if S ⊂ T , and σ-subadditive, i.e. if (S n ) is
a countable family of elements of E then
µ (∪ n S n ) ≤ ∑ n
µ (S n )
Probability measures are continuous from below and from above, in the following sense: if S n ↑ S
then µ (S n ) ↑ µ (S), and if S n ↓ S then µ (S n ) ↓ µ (S). Both continuity properties are equivalent,
and indeed a simple argument shows that they are equivalent to continuity from above at ∅: if
S n ↓ ∅ then µ (S n ) ↓ 0. Moreover, continuity is equivalent to σ-aditivity if the set function µ is
only assumed (finitely) additive.
A subset E ⊂ X has zero measure if it is contained in a measurable set S ∈ E with µ (S) = 0.
If any set with zero measure belongs to E, then the measure space (X, E, µ) is said complete. Any
measure space can be canonically completed, extending the measure to the σ-algebra E made of
E and of subsets of zero measure. A property (like continuity of a function, or convergence of a
sequence of functions) holds µ-a.e. (“almost everywhere” with respect to the measure µ) if the set
of points of X where it does not hold has zero measure.
Construction of probability measures. Measures are never ”explicitely” given as functions
on a σ-algebra. A set function µ : P(X) → [0, ∞] is an exterior measure if it is monotone,
σ-subadditive, and if µ (∅) = 0. It happens that, given an exterior measure µ, the family of
µ-measurable sets, defined as
E = {E ⊂ X such that µ(S) = µ(S ∩ E) + µ(S ∩ E c ) for any S ⊂ X}
is a σ-algebra, and that µ is a complete measure if restricted on E (the proof is quite long and
delicate, but the only idea it uses is the following: in order to check that E ∈ E it is indeed sufficient,
by virtue of monotonicity and subadditivity of µ, to check that µ(S) ≥ µ(S ∩ E) + µ(S ∩ E c ) for
any S ⊂ X). A strategy to construct interesting measures on uncountable spaces is: start with an