Number Theory: An Introduction to Mathematics

470 XI Uniform Distribution and Ergodic Theory

It should be noticed that the preceding proof simplifies if the functionfis bounded.
In Birkhoff’s original formulation the functionfwas the indicator functionχBof an
arbitrary setB∈B. In this case the theorem says that, ifvn(x)is the number ofk<n
for whichTkx∈B, then limn→∞vn(x)/nexists for almost allx∈X. That is, ‘almost
every point has an average sojourn time in any measurable set’.
A measure-preserving transformationTof the probability space(X,B,μ)is said
to beergodicif, for everyB∈BwithT−^1 B=B, eitherμ(B)=0orμ(B)=1.
Part (ii) of the next proposition says that this is the case if and only if ‘time means and
space means are equal’.

Proposition 18Let T be a measure-preserving transformation of theprobability
space(X,B,μ). Then T is ergodic if and only if one of the following equivalent
properties holds:

(i)if f∈L(X,B,μ)satisfies f(Tx)=f(x)almost everywhere, then f is constant

almost everywhere;

(ii)if f∈L(X,B,μ)then, for almost all x∈X,

n−^1

n∑− 1

k= 0

f(Tkx)→

∫

X

fdμ as n→∞;

(iii)if A,B∈B,then

n−^1

n∑− 1

k= 0

μ(T−kA∩B)→μ(A)μ(B) as n→∞;

(iv)if C∈Bandμ(C)> 0 ,thenμ(

⋃

n≥ 1 T

−nC)= 1 ;

(v)if A,B∈Bandμ(A)> 0 ,μ(B)> 0 ,thenμ(T−nA∩B)> 0 for some n> 0.

Proof Suppose first thatTis ergodic and letf∈L(X,B,μ)satisfyf(Tx)=f(x)
a.e. Put

f ̄(x)= lim

n→∞

n−^1

n∑− 1

k= 0

f(Tkx).

Thenf ̄(Tx)=f ̄(x)for everyx∈Xandf ̄(x)=f(x)a.e. For anyα∈R,let

Aα={x∈X:f ̄(x)<α}.

Thenμ(Aα)=0or1,sinceT−^1 Aα=AαandTis ergodic. Sinceμ(Aα)is a nonde-
creasing function ofαandμ(Aα)→0asα→−∞,μ(Aα)→1asα→+∞,there
existsβ∈Rsuch thatμ(Aα)=0forα<βandμ(Aα)=1forα>β. It follows
thatμ(Aβ)=0andμ(Bβ)=1, where

Bβ={x∈X:f ̄(x)≤β}.

Hencef(x)=βa.e. and (i) holds.