302 CHAPTER 12. THE CONSTRUCTION OF MEASURES

Definition 12.8.3 A measurable function f , is said to be invariant if

f (T ω) = f (ω) .

A set, A ∈F is said to be invariant if XA is an invariant function. Thus a set is invariantif and only if T−1A = A. (XA (T ω) = XT−1(A) (ω) so to say that XA is invariant is to saythat T−1A = A.)

The following theorem, the individual ergodic theorem, is the main result. DefineT 0 (ω) = ω . Let

Sn f (ω)≡n

∑k=1

f(

T k−1ω

), S0 f (ω)≡ 0.

Also define the following maximal type function M∞ f (ω)

M∞ f (ω)≡ sup{Sk f (ω) : 0≤ k} (12.8.21)

and letMn f (ω)≡ sup{Sk f (ω) : 0≤ k ≤ n} (12.8.22)

Then one can prove the following interesting lemma.

Lemma 12.8.4 Let f ∈ L1 (µ) where f has real values. Then∫[M∞ f>0] f dµ ≥ 0.

Proof: First note that Mn f (ω) ≥ 0 for all n and ω . This follows easily from theobservation that by definition, S0 f (ω) = 0 and so Mn f (ω) is at least as large. There iscertainly something to show here because the integrand is not known to be nonnegative.The integral involves f not M∞ f .

Let T ∗h≡ h◦T . Thus T ∗ is linear and maps measurable functions to measurable func-tions by Lemma 12.8.1. It is also clear that if h≥ 0, then T ∗h≥ 0 also. Therefore, for largek ≤ n,

Sk f (ω) ≡k

∑j=1

f(T j−1

ω)= f (ω)+

k

∑j=2

f(T j−1

ω)

= f (ω)+T ∗k−1

∑j=1

f(T j−1

ω)

(factored out T ∗)

= f (ω)+T ∗Sk−1 f (ω)≤ f (ω)+T ∗Mn f

and so, taking the supremum for k ≤ n,

Mn f (ω)≤ f (ω)+T ∗Mn f (ω) .

Now since Mn f ≥ 0, ∫Ω

Mn f (ω)dµ =∫[Mn f>0]

Mn f (ω)dµ

≤∫[Mn f>0]

f (ω)dµ +∫

Ω

T ∗Mn f (ω)dµ