My title - Departamento de Matemática da Universidade do Minho

10 STATISTICAL DESCRIPTION OF ORBITS 64
exterior measure (it is very easy to produce exterior measures, for example by means of variational
principles) and then check that the σ-algebra of measurable sets is sufficiently big for our purpose.
The idea of Carathéodory is the following. A probability measure on an algebra A of subsets
of X is an additive function m : A → [0, 1] such that m (∅) = 0, m (X) = 1, and such that A n ↓ ∅
implies m (A n ) ↓ 0. Given a probability measure m on a algebra A , the recipe
{∑ }
µ (S) = inf m(An ) with S ⊂ ∪ n A n e A n ∈ A
defines an exterior measure on P(X), hence the above construction produces a measure µ on the
σ-algebra of µ-measurable sets, which contains A and so contains σ (A). One then checks that
µ (A) = m (A) for any A ∈ A, so that µ is an “extension” of the measure m. Carathéodory’s
extension theorem is then stated in the following form:
Carathéodory’s extension theorem . Given a probability measure m on a algebra A of
subsets of X, there exists a unique measure µ on σ (A) which extends m.
The following corollary of Carathéodory’s theorem is also useful, for example when trying to
prove that some event has a definite probability.
Approximation theorem . Let (X, E, µ) be a probability space, and let A be an algebra of
subsets of X such that σ (A) = E. Then, for any A ∈ E and any ε > 0, we can find a A ε ∈ A
such that
µ (A∆A ε ) < ε.
Indeed, one easily sees that the family C = {A ∈ E s.t. ∀ε > 0 ∃A ε ∈ A s.t. µ (A∆A ε ) < ε} is
a σ-algebra. Since A is obviously contained in C, this implies that E =σ (A) ⊂ C ⊂ E.
Lebesgue measure. The collection I of intervals of the real line is a semi-algebra, i.e. the
intersection of two elements of I is in I and the complement of an element of I is a union of
elements of I. The function m : I → [0, ∞], defined as m ([a, b]) = |b − a| if a e b are finite, and
∞ if the interval is unbounded, is monotone and gives value zero to the empty set. Postulating
additivity, the function m extends to a measure on the algebra A made of disjoint unions of
elements of I (this is not trivial!, the proof uses the Heine-Borel theorem about compact subsets
of the real line). The function µ : P(R) → [0, ∞], defined as
{∑ }
µ (E) = inf m(Cn ) with E ⊂ ∪ n C n e C n ∈ A
is then an exterior measure on the real line. The σ-algebra L of µ-measurable sets, called Lebesgue
σ-algebra, contains the Borel sets, because it contains the intervals. The restriction l = µ | L , as
well as µ ∣ ∣ B(R) , is called Lebesgue measure.
Observe that Lebesgue measure on the real line is not a probability measure, having infinite
mass. Nevertheless, one can easily define probability measures on bounded intervals taking
normalized restrictions of Lebesgue measure. For example, take X = [0, 1], and E = B (X) =
{X ∩ B with B ∈ B (R)}, the Borel subsets of the interval. The restriction of l to E is a probability
measure, called Lebesgue measure on the unit interval.
The very same construction works in R n , starting with the semi-algebra of “rectangles” measured
by the “euclidean volume”, and produces a measure l on B (R n ), also called Lebesgue measure.
Lebesgue measure is the unique measure over the Borel sets of the euclidean space which is invariant
under traslations, i.e. l (λ + B) = l (B) for any λ ∈ R n and any Borel set B, and which is
normalized to give measure one to the unit square, i.e. l ([0, 1] n ) = 1.
The axiom of choice allows one to “give examples” of subsets wich are not Lebesgue-measurable
(for example, the set made of one point for each orbit of an irrational rotation of the circle).
The following result is useful (see [Mat95] for a proof). Below, B ε (x) = {y ∈ R n s.t. ‖x −
y‖ 2 < ε} denotes the open ball of radius ε > 0 and center x ∈ R n w.r.t. the Euclidean distance
‖x − y‖ 2 2 = ∑ n
i=1 |x i − y i | 2 .