478 XI Uniform Distribution and Ergodic Theory
We m a y d e fi n e t h egeneral cylinder set Cia 11 ......iakk,wherei 1 ,...,ikare distinct inte-
gers, to be the set of allx∈Xsuch that
xi 1 =a 1 ,...,xik=ak.In particular,Cia=σ−i[a] and henceμ(Cia)=pa. It follows by induction onkthat
μ(Cia 11 ......iakk)=pa 1 ···pak.Proposition 24For any given positive numbers p 1 ,...,prwith sum 1 , the two-sided
Bernoulli shift Bp 1 ,...,pris ergodic.
Proof SupposeB∈Bandσ−^1 B=B.Foranyε>0 there exists a setC∈Csuch
that
μ(B∆C)=μ(B\C)+μ(C\B)<ε.Then
|μ(B)−μ(C)|=|μ(C∩B)+μ(B\C)−μ(C∩B)−μ(C\B)|
≤μ(B\C)+μ(C\B)<εand hence
|μ(B)^2 −μ(C)^2 |={μ(C)+μ(B)}|μ(B)−μ(C)|< 2 ε.We may suppose thatCis the union of finitely many special cylinder sets of orderm.
Since
σ−n[a−m,...,am]=C−a−mm+,...,n,...,amm+n,forn> 2 mwe have
[a−′m,...,am′]∩σ−n[a−m,...,am]=C
a−′m,...,am′,a−m,...,am
−m,...,m,−m+n,...,m+n,and hence
μ([a−′m,...,am′]∩σ−n[a−m,...,am])=pa′−m···pa′mpa−m···pam,
=μ([a′−m,...,am′])μ([a−m,...,am]).It follows that ifn> 2 m,then
μ(C∩σ−nC)=μ(C)^2.But
μ(B\(C∩σ−nC))≤ 2 μ(B\C),