Number Theory: An Introduction to Mathematics

4 Applications 477

B=[a−m,...,am]∪···∪[a′−m,...,a′m],

we define

μm(B)=pa−m···pam+···+pa′−m···pam′.

Thenμm(X)=1,μm(B)≥0foreveryB∈Cm,and

μm(B∪C)=μm(B)+μm(C)ifB,C∈CmandB∩C=∅.

Every union of cylinder sets of ordermis also a union of cylinder sets of order
m+1, since

[a−m,...,am]=∪a,a′∈A[a,a−m,...,am,a′].

ThusCm⊆Cm+ 1. Moreoverμm+ 1 continuesμm,since

μm+ 1 ([a−m,...,am])=

∑r

j,j′= 1

pjpj′pa−m···pam

=μm([a−m,...,am])

(∑r

j= 1

pj

)(∑r

j′= 1

pj′

)

=μm([a−m,...,am]).

Letμdenote the continuation of allμmtoC=C 0 ∪C 1 ∪....IfB,C∈C,then
B,C∈Cmfor somem. Hence, for givenC∈C, there are only finitely many distinct
B∈Csuch thatB⊆C. Consequently, ifCis the union of a sequence of disjoint sets
Cn ∈ C(n = 1 , 2 ,...),thenCn =∅for all largenandμ(C)=

∑

n≥ 1 μ(Cn).
It follows, by a construction due to Carath ́eodory (1914), thatμcan be uniquely
extended to theσ-algebraBof subsets ofXgenerated byCso that (X,B,μ)isa
probability space. For anyε>0 there exists, for eachB∈B,someC∈Csuch that
μ(B∆C)<ε.
Thetwo-sided Bernoulli shift Bp 1 ,...,pris the mapσ:X→Xdefined byσx=x′,
wherexi′=xi+ 1 for everyi∈Z. It is a measure-preserving transformation of the
probability space (X,B,μ), since

σ−^1 [a−m,...,am]=∪a,a′∈A[a,a′,a−m,...,am]

and hence

μ(σ−^1 [a−m,...,am])=

∑r

j,j′= 1

pjpj′pa−m···pam

=

∑r

j,j′= 1

pjpj′μ([a−m,...,am])=μ([a−m,...,am]).

The Bernoulli shiftB 1 / 2 , 1 / 2 is a model for the random process consisting of bi-infinite
sequences of coin-tossings.