102 Hopf, On the ergodtc theorem for positive linear operators.

of their reasoning. Our subsequent proof is very simple. Little touches have been applied in other places too to simplify the presentation. It must be said, however, that the principal ideas are those of Chacon and Ornstein.

The proof makes use of the lattice operations /->/+, f-^f~ which carry L^ into L^,

( 2 ) /+ = max(/,0), /-= - min (/, 0).

There holds

( 3 ) f-i^-i-, 1/1=/+ + /-.

/ + , /~ are both ^ 0 but never both > 0. Therefore

( 4 ) /=/2-/1, A^o ^ /+^/„/-^/,.

From Tf = Г/+— Г/-, from the positivity of T and from (4) it follows that

( 5 ) (Tfy < Tf^, (Tf)- ^Tf-, \Tf\<T\f\.

The assumption that | Г | ^ 1 is equivalent to the statement that

( 6 ) (p^O ^ jT(p<j(p,

The proof of the ergodic theorem is carried through in several steps the first of which is the proof of the

Basic lemma 1. // f iL^ and if

n—l

sup j; T^f >0

n>0 0

holds in each point of a set A (belonging to the giçen a-field) then there exist, to each e> 0, functions h ^ L^, (p ^ L^ such that

a ) h- ^ /-,

ß ) h=f+ T<p~<p, 9)^0,

y ) fh ^ ff, Ô) fh- <e.

A

Proof . We define a sequence of functions A, €L^, i ^ 0, h^ = f, such that A^+i is obtained from A, on applying T only to its positive part,

( 7 ) K^,^Th+-h-,ho=f.

We show that each h = h^ has the properties a) — y) and that à) is satisfied for ciently large i. From (7), the positivity of T and from (4),

( 8 ) K^i^K

and this implies a). (7) may be written

A , + i=A , + Tht—ht^ Hence, by summation,

( 9 ) Аг=/+ T(p,__, — (p,_^,(p,=2;ht,

0

which proves ß). But ^8) in turn implies y) by virtue of (6). From 9?^— ç)^_i = h^ ^ A^ and from the first equation (9),

( Pt^f + T(p,_^,

I > 0. As 7* is order-preserving this implies, by induction, that

0