12.8. THE ERGODIC THEOREM 303

=∫[Mn f>0]

f (ω)dµ +∫

Ω

Mn f (ω)dµ

by Lemma 12.8.2. It follows that ∫[Mn f>0]

f (ω)dµ ≥ 0

for each n. Also, since Mn f (ω)→M∞ f (ω) , the following pointwise convergence holds.

X[Mn f>0] (ω) f (ω)→X[M∞ f>0] (ω) f (ω)

Since f is in L1, the dominated convergence theorem implies∫[M∞ f>0]

f (ω)dµ = limn→∞

∫[Mn f>0]

f (ω)dµ ≥ 0.

Theorem 12.8.5 Let (Ω,F ,µ) be a probability space and let T : Ω→ Ω satisfy 12.8.18and 12.8.19, T−1 is measure preserving and T−1 maps F to F and T is one to one. Thenif f ∈ L1 (Ω) having real or complex values and

Sn f (ω)≡n

∑k=1

f(

T k−1ω

), S0 f (ω)≡ 0, (12.8.23)

it follows there exists a set of measure zero N, and an invariant function g such that for allω /∈ N,

limn→∞

1n

Sn f (ω) = g(ω) . (12.8.24)

and alsolimn→∞

1n

Sn f = g in L1 (Ω)

Proof: To begin with, we assume f has real values. Now if A is an invariant set,XA (T mω) = XA (ω) and so

Sn (XA f )(ω)≡n

∑k=1

f(

T k−1ω

)XA

(T k−1

ω

)=

n

∑k=1

f(

T k−1ω

)XA (ω)

= XA (ω)n

∑k=1

f(

T k−1ω

)= XA (ω)Sn f (ω) .

Therefore, for such an invariant set,

Mn (XA f )(ω) = XA (ω)Mn f (ω) , M∞ (XA f )(ω) = XA (ω)M∞ f (ω) . (12.8.25)

Let −∞ < a < b < ∞ and define

Nab ≡[−∞ < lim inf

n→∞

1n

Sn f (ω)< a < b < lim supn→∞

1n

Sn f (ω)< ∞

](12.8.26)