A NNALES DE L’I. H. P., SECTION B
T RUMAN B EWLEY
Extension of the Birkhoff and von Neumann ergodic
theorems to semigroup actions
Annales de l’I. H. P., section B, tome 7, no 4 (1971), p. 283-291
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Ann. Inst. Henri Poincaré,
Section B :
Vol. VII, n° 4, 1971, p. 283-291.
Calcul des Probabilités et
Statistique.
Extension of the Birkhoff and von Neumann
ergodic theorems to semigroup actions (*)
Truman BEWLEY
In 1967, A. A. Tempelman announced generalizations of the Birkhoff
and von Neumann ergodic theorems [6]. This paper supplies proofs of
results similar to Tempelman’s. The main arguments are drawn from
Calderon’s paper [l ]. The author has also had the benefit of reading
Mrs. J. Chatard’s work on the same problem [2].
PRELIMINARIES
Let (M, M, ) and (G, J, y) be complete measure spaces,
where is
~-finite. Assume that G is a semigroup with product indicated by juxtaposition and that there is a map (x, m) - x(m) from G x M to M, measurable with respect to G x ~ and such that x(y(m))
xy(m) for all
x, y E G and mE M. Assume that
for all xeG and
where
Finally, assume that for all
x E G and
xE and Ex are measurable, y(xE)
y(E) y(Ex), and
that
and Ex -1 are measurable, where x -1 E = ~ y e G : xy e
E ~ and
=
=
If
x E
G and E and D
are
in
=
~, then
(*) This work was written for a seminar in ergodic theory organized by Professor A. AvEZ.
The seminar was supported by contract number AEC
AT(11-1)-34.
The author would like to express his gratitude to Professor AvEZ for
encouragement
and hepful comments.
284
so
TRUMAN BEWLEY
that
where xD is the characteristic function of D.
grable function
and
f
on
Therefore, if f is
any inte-
G,
similarly,
If f is any function
is any nonnegative
on
M and
integrable
x E
G, define fx by fx(m)
on M,
=
f (xm).
Let A~ be a sequence of measurable subsets of G such that
for all n. We shall use the following conditions on the A~.
1.
n
m
.
II.
lim
implies A~, c A~;
xAJ=
III.
A.,x)
lim
denotes
=
0, for all
x e
oo
G, where A
symmetric difference;
for each k and n,
is measurable and lim
=
n
IV.
If
function
there exists K
A~
>
=
1 such that
{xeG :
yx
e
K(AJ
A~
for
some
0 ; and
for all n, where
ye
Let B be a real or complex Banach space with norm [ [ . [ and dual B*.
If /~eB* and
~(b) will be denoted by ~’h. If (N, v) is a measure
if
and
1
~, LBp(N, v) will denote the set of equivalence classes
space
p
of functions f : M -~ B such that f is the limit in measure of simple
functions and
L: is a Banach space, and if (N, v) is a-finite, its dual is Lq *, where
if p
=
1,
and - + - = 1 otherwise.
q
P
285
BIRKHOFF AND VON NEUMANN ERGODIC THEOREMS TO SEMIGROUP ACTIONS
If
we
imply
choose g
so
>
0
a.
e., then
(1) and Fubini’s theorem
that for almost every m,
exists.
that
It follows that for almost every m,
Applying
Fubini’s theorem
Hence, the map
and the
integral
is
again
and
f .~(m) E
using (2),
we
y
IAn) and
hence
obtain
from A" to L: is integrable.in the sense of Pettis [5],
equal
to
fx(m)dy(x)
almost
everywhere. Define
Clearly, nn is a continuous linear operator of norm less than or equal to one.
THE ERGODIC THEOREMS
THEOREM 1 (von Neumann’s Mean Ergodic Theorem).
If the
II
and
if
1
o0
or
1
and
satisfy
p
if p
Jl(M) oo, then there is n( f) E
such that
n( f) I ( p - 0 and such that
n( f)
-
=
lim
for all
xE
G.
L:
=
If p
1,
is the
of
Lp
Ip along Mp, where Ip is the subspace of invariant functions and M:
the closed
=
=
03C0
projection
onto
is
An
subspace generated by { fx - f :
ANN. INST.
POINCARE, B-VII-4
20
286
TRUMAN BEWLEY
THEOREM 2
(Wiener-Calderon Dominated Convergence Theorem).
Suppose that the A" satisfy I, III, and IV. If f is a nonnegative integrable
-
function and if for
0,
a >
then
THEOREM 3 (BirkhofFs Individual Ergodic Theorem). If the A" satisfy
I-IV and if f E
where 1
oo, then nn(f) converges almost everyp
where. If 1 p
o0 or if p
1 and
oo, then nn(f) converges
almost everywhere to the n( f) of Theorem 1.
L:,
=
PROOF OF THEOREM 1
LEMMA 1.
is
-
If
f ~ LBp
and 1
~, then
p
C( f ) = {fx:
weakly compact.
Proof.
If p
>
1
let1-q + 1P - 1.
If p
1let q
=
=
~.
C (f) is weakly
compact if for each sequence xm E G and each sequence ~,n E Lq
that [ ~,n ~
1 for all n, lim lim
m
n
~.n ’fxm
=
m
lim lim
n
m
~,n ’fxm
m
*
such
whenever each
limit exists [4, p. 159].
Let E > 0 and choose a simple function g E Lp such that( f - g
E.
for
all
so
n
and
that
we
a
c
I ~n
m,
I I I g I Ip
may, by diagonal process,
choose a subsequence gx mk such that for each n, lim 03B n.gxm, ~
an. We
may
that an
such that lim
assume
gxmk
~
a.
Similarly,
03BBnl. gxmk
=
we
may choose
ck for each k.
Again,
that Ck ~ C. Since (
~Bp I ~B*q~fxm all m, n, it suffices to show that a
c.
Since the ~,n are uniformly bounded, the sequence
a
we
subsequence
may
gx,.,.,
~Bp
assume
E
for
=
has
a
weak star
s
limit
point, 03BB0.
g
for all x E G, {
xmk has a subnet
for each i. Then,
=
hibi,
x E
where
Lp and
Since ) )
G} is weakly relatively compact for each i. Hence,
such that
a
B.
=
~,o ~
=
c.
Q.
converges
E. D.
weakly
to some
hio
287
BIRKHOFF AND VON NEUMANN ERGODIC THEOREMS TO SEMIGROUP ACTIONS
PROOF OF THEOREM 1.
C( f ) is weakly compact,
G > 0, there are
with
2014
has
~03BD~Bp
a
G
> 1.
weak cluster point
i = 1,
and a;,
Since
Suppose that p
~( f ). Given
...,
m, with
such that
Hence,
For every
so
that
x E
G,
n( f) is invariant and
E
it follows that
=
n( f) for all
n.
Hence,
and since for all xeG,
~Bp
=
0,
The case p
1 follows from the case p
2, since the nn are uniformly
bounded on LB1 and since, if Jl(M)
oo, LB2 is BB1-dense in LB2 and the
~. ~B2-topology is stronger than the LB-topology. Q. E. D.
=
=
PROOF OF THEOREM 2
The key step of the proof is the Wiener-Calderon
made in proving Lemma 1 below.
Let « B » denote set theoretic difference.
covering argument
288
TRUMAN BEWLEY
LEMMA 1.
Let h be
-
a
real-valued
Suppose that
for
some
f
Proof
form Akx,
1,
=
...,
K, where
a >
y-integrable
and the
Let ~k be a maximal collection of
such that
disjoint subsets
of the
be a maximal collection of
+ 1, where k > i > 1, let
where
A;x
x, E An and such that
sets of
A;x
mutually disjoint
are
and
are
disjoint
from every set in
k
for i + 1
j
k.
Let M
Let
and
k(x)
0.
Then
x E Ai-l Ajx’ c
h > 0 on N and
=
~~
U
A;x E
x E An and
Therefore, there exists
But then
on
0.
-
Given ~l~
the form
function defined
Let N
for some i
AkAnBu N. Suppose
=
=
x
e N
1, ..., k,
such that Atx n Ajx’~ 03C6.
This contradicts xeN. Hence,
with j~i
289
BIRKHOFF AND VON NEUMANN ERGODIC THEOREMS TO SEMI GROUP ACTIONS
LEMMA 2.
/(~) ~
0
or
/(~) ~ 2014 2014
K
and for
some
f
=
1,
...,
k,
0.
Then,
Proof - Let n be a
be defined by
Let
either
and that for each
Suppose that
-
positive integer and for each
M’= { m E M :
rem,
=
Lemma 1.
Applying
m E M, let
hm : AkAn
is y-integrable on AkAn }. By Fubini’s TheoM’, hm satisfies the assumptions of
0.
For each m E
Hence,
Fubini’s Theorem,
have
we
or
Let n -
oo
and
PROOF
OF
THEOREM 2. It is sufficient to prove that
where
Jl(F)
oo
apply
III.
Q.
E. D.
and
Let
Since h satisfies the
R
assumptions
of Lemma 3,
0.
Q. E.
D.
290
TRUMAN BEWLEY
PROOF OF THEOREM 3
oo.
Suppose at first that if p 1,
~( f ) be the limit defined by the Mean Convergence Theorem.
~n( f - ~( f )) ~c"( f ) - ~c( f ), one may suppose that ~( f ) 0. We
that ~"( f ) -~ 0 a. e.
=
Let
=
Let E
>
k such
0.
=
Choose f b bounded and such that
that !! ~cx( f b) - ~( fb) ~~B
Clearly, [ ) H ~p
If p
Since
=
E
is
Then, f = H
+
E.3
Choose
G, where
E.
converges to
Therefore, if 03B4
E.3
[ f b - f ~ ~B
Since
show
>
almost
zero
everywhere since, for almost
0,
1, by Theorem 2
we
arbitrarily small,
obtain
we
obtain lim
n
=
0
a. e.
every m,
291
BIRKHOFF AND VON NEUMANN ERGODIC THEOREMS TO SEMI GROUP ACTIONS
Furthermore,
so
that
almost
by
Theorem
2,
everywhere.
1
oo as in the case p
that
E
E
set
a
We
call
and prove that
converges almost everywhere.
invariant if xE c E, Vx E G. It is possible to find a sequence of invariant
sets, Ik, of finite measure and such that if I is invariant and measurable
=00.
0 or
n
and if
By what we
U Ik) 0, then either
We
the
now remove
assumption
=
=
=
k
already proved,
have
Let
The
G >
0 and let
measure
fb
converges
on
each
Ik and
hence
on
I =
U Ik.
k
be
a
of the first set
bounded function such
on
the
right
show that the second set is invariant.
it must have measure zero. Q. E. D.
to
that !!
is bounded
by
G.
~~6 .
Since it is bounded
It is easy
K
by ~ [
[ [t
BIBLIOGRAPHY
[5]
P. CALDERON, A General Ergodic Theorem. Annals of Mathematics, t. 58, 1953,
p. 182-191.
J. CHATARD, « Application des propriétés de moyenne d’un groupe localement compact
à la théorie ergodique », Université de Paris (mineographed).
N. DUNFORD and J. SCHWARTZ, Linear Operator, Part I. Interscience, 1966.
J. L. KELLY, I. NAMIOKA and coauthors, Linear Topological Spaces. Van Nostrand,
1963.
B. J. PETTIS, On Integration on Vector Spaces. Trans. Amer. Math. Soc., t. 44, 1938,
[6]
A. A. TEMPELMAN,
[1] A.
[2]
[3]
[4]
p. 277-304.
Ergodic
Theorems for General
Dynamic Systems.
8, 1967, p. 1213-1216.
[7] Norbert WIENER, The Ergodic Theorem. Duke Math. Journal,
Dokl.,
Soviet Math.
t.
t.
5, 1934, p. 1-18.
Manuscrit reçu le 9
juin 1971.