Number Theory: An Introduction to Mathematics

466 XI Uniform Distribution and Ergodic Theory

Thenμis said to be aprobability measureand the triple(X,B,μ)is said to be a
probabilityspace.
It is easily seen that the definition implies
(i)μ(∅)=0,
(ii)μ(Bc)= 1 −μ(B),
(iii)μ(A)≤μ(B)ifA,B∈BandA⊆B,
(iv)μ(Bn)→μ(B)if(Bn)is a sequence of sets inBsuch thatB 1 ⊇B 2 ⊇···and
B=

⋂

nBn.
If a property of points in a probability space(X,B,μ)holds for allx∈B,where
B∈Bandμ(B)=1, then the property is said to hold for (μ-)almost all x∈X,or
simplyalmost everywhere(a.e.).
A function f : X → R is measurableif, for everyα ∈ R,theset
{x ∈ X : f(x)<α}is inB.Letf : X →[0,∞)be measurable and for any
partitionPofXinto finitely many pairwise disjoint setsB 1 ,...,Bn∈B, put

LP(f)=

∑n

k= 1

fkμ(Bk),

wherefk=inf{f(x):x∈Bk}. We say thatfisintegrableif
∫

X

fdμ:=sup

P

LP(f)<∞.

The set of all measurable functionsf :X→Rsuch that|f|is integrable is denoted
byL(X,B,μ).
AmapT:X→Xis said to be ameasure-preserving transformationof the prob-
ability space(X,B,μ)if, for everyB∈B,thesetT−^1 B={x∈X:Tx∈B}is
again inBandμ(T−^1 B)=μ(B). This is equivalent toμ(TB)=μ(B)for every
B∈Bif the measure-preserving transformationTisinvertible,i.e.ifTis bijective
andTB∈Bfor everyB∈B. However, we do not wish to restrict attention to the
invertible case. Several important examples of measure-preserving transformations of
probability spaces will be given in the next section.
Birkhoff ’s ergodic theorem, which is also known as the ‘individual’ or ‘pointwise’
ergodic theorem, has the following statement:

Theorem 17Let T be a measure-preserving transformation of the probability space
(X,B,μ).If f∈L(X,B,μ)then, for almost all x∈X , the limit

f∗(x)= lim

n→∞

n−^1

n∑− 1

k= 0

f(Tkx)

exists and f∗(Tx)=f∗(x). Moreover, f∗∈L(X,B,μ)and

∫

Xf

∗dμ=∫

Xfdμ.

Proof It is sufficient to prove the theorem for nonnegative functions, since we can
writef=f+−f−,where

f+(x)=max{f(x), 0 }, f−(x)=max{−f(x), 0 },

andf+,f−∈L(X,B,μ).