494 CHAPTER 18. TOPOLOGICAL VECTOR SPACES

18.4 Mean Ergodic TheoremThe following theorem is called the mean ergodic theorem.

Theorem 18.4.1 Let (Ω,S ,µ) be a finite measure space and let T : Ω→Ω satisfy

T−1 (E) ∈S ,T (E) ∈S

for all E ∈S . Also suppose for all positive integers, n, that

µ(T−n (E)

)≤ Kµ (E) .

For f ∈ Lp (Ω), and p > 1, letT ∗ f ≡ f ◦T. (18.4.9)

Then T ∗ ∈L (Lp (Ω) ,Lp (Ω)) , the continuous linear mappings form Lp (Ω) to itself with

||T ∗n|| ≤ K1/p. (18.4.10)

Defining An ∈L (Lp (Ω) ,Lp (Ω)) by

An ≡1n

n−1

∑k=0

T ∗k,

there exists A ∈L (Lp (Ω) ,Lp (Ω)) such that for all f ∈ Lp (Ω) ,

An f → A f weakly (18.4.11)

and A is a projection, A2 = A, onto the space of all f ∈ Lp (Ω) such that T ∗ f = f . (Theinvariant functions.) The norm of A satisfies

||A|| ≤ K1/p. (18.4.12)

Proof: To begin with, it follows from simple considerations that∫|XA (T n (ω))|p dµ =

∫ ∣∣XT−n(A) (ω)∣∣p dµ = µ

(T−n (A)

)≤ Kµ (A)

Hence∥T ∗n (XA)∥ ≤ K1/p

µ (A)1/p = K1/p ∥XA∥Lp

Next suppose you have a simple function s(ω) = ∑nk=1 XAi (ω)ci where we assume the Ai

are disjoint. From the above,

∫ ∣∣∣∣∣ n

∑k=1

XAi (Tm (ω))ci

∣∣∣∣∣p

dµ =∫ n

∑k=1

XAi (Tm (ω))p |ci|p dµ

≤n

∑k=1

Kµ (Ai) |ci|p = K∫|s|p dµ