Number Theory: An Introduction to Mathematics

3 Birkhoff’s Ergodic Theorem 469

since the measure-preserving nature ofTimplies that, for anyg∈L(X,B,μ),

∫

X

g(Tx)dμ(x)=

∫

X

g(x)dμ(x).

Since
∫

X

f ̃dμ≤

∫

X

fdμ+

∫

X\EN

Mdμ≤

∫

X

fdμ+ε,

it follows that
∫

X

f ̄Mdμ≤

∫

X

fdμ+ 2 ε+NM/L.

SinceLmay be chosen arbitrarily large and thenεarbitrarily small, we conclude that

∫

X

f ̄Mdμ≤

∫

X

fdμ.

Now lettingM→∞, we obtain

∫

X

f ̄dμ≤

∫

X

fdμ.

The proof that

∫

X

fdμ≤

∫

X

fdμ

is similar. Givenε>0, there exists for eachx∈Xa positive integernsuch that

n−^1

n∑− 1

k= 0

f(Tkx)≤f(x)+ε. (∗∗)

IfFnis the set of allx∈Xfor which (∗∗) holds and ifEn=

⋃n
k= 1 Fk, we can choose
Nso large that

∫

X\EN

fdμ<ε.

Put

f ̃(x)=f(x) ifx∈EN,

=0ifx∈/EN.

Letτ(x)be the least positive integernfor which (∗∗) holds ifx∈EN,andτ(x)= 1
otherwise. The proof now goes through in the same way as before.