Number Theory: An Introduction to Mathematics

3 Birkhoff’s Ergodic Theorem 465

actual state of motion, will, sooner or later, pass through every phase which is con-
sistent with the equation of energy”. The wordergodic, coined by Boltzmann (1884),
was a composite of the Greek words for ‘energy’ and ‘path’. It was recognized by
Poincar ́e (1894) that it was too much to ask that a path pass through every state on the
same energy surface as its initial state, and hesuggested instead that it pass arbitrarily
close to every such state. Moreover, he observed that it would still be necessary to
exclude certain exceptional initial states.
A breakthrough came with the work of G.D. Birkhoff (1931), who showed that
Lebesgue measure was the appropriate tool for treating the problem. He established a
deep and general result which says that, apart from a set of initial states of measure
zero, there is a definite limiting value for the proportion of time which a path spends in
any given measurable subsetBof an energy surfaceX. The proper formulation for the
ergodic hypothesis was then that this limiting value should coincide with the ratio of
themeasureofBto that ofX, i.e. that ‘the paths through almost all initial states should
be uniformly distributed over arbitrary measurable sets’. It was not difficult to deduce
that this was the case if and only if ‘any invariant measurable subset ofXeither had
measure zero or had the same measure asX’.
Birkhoff proved his theorem in the framework of classical mechanics and forflows
with continuous time. We will prove his theorem in the abstract setting of probability
spaces and forcascadeswith discrete time. The abstract formulation makes possible
other applications, for which continuous time is not appropriate.
LetBbe aσ-algebraof subsets of a given setX, i.e. a nonempty family of subsets
ofXsuch that

(B1)the complement of any set inBis again a set inB,
(B2)the union of any finite or countable collection of sets inBis again a set inB.

It follows thatX∈B,sinceB∈BimpliesBc:=X\B∈BandX=B∪Bc.
Hence also∅=Xc ∈B. Furthermore, the intersection of any finite or countable
collection of sets inBis again a set inB,since

⋂

nBn= X\

(⋃

nB

c

n

)

. Hence if
A,B∈B,then

B\A=B∩Ac∈B

and thesymmetric difference

A∆B:=(B\A)∪(A\B)∈B.

The family of all subsets ofXis certainly aσ-algebra. Furthermore, the intersec-
tion of any collection ofσ-algebras is again aσ-algebra. It follows that, for any family
Aof subsets ofX,thereisaσ-algebraσ(A)which containsA and is contained
in everyσ-algebra which containsA. We callσ(A)theσ-algebra of subsets ofX
generatedbyA.
SupposeBis aσ-algebra of subsets ofXand a functionμ:B→Ris defined
such that

(Pr1)μ(B)≥0foreveryB∈B,
(Pr2)μ(X)=1,
(Pr3)if(Bn)is a sequence of pairwise disjoint sets in B,thenμ

(⋃

nBn

)

∑ =

nμ(Bn).