Internat. J. Math. & Math. Sci.
VOL. 15 NO.
(1992) 195-198
195
RESEARCH NOTES
A NOTE ON CONSERVATIVE MEASURES ON SEMIGROUPS
N.A. TSERPES
Department of Mathematics
University of Patra
Patra, Greece
(Received June 6, 1990)
Consider (S,B,p) the measure space where S is a topological metric semi-
ABSTRACT.
a countably additive bounded Borel measure. Call
group and
right translations
ix:
sx, x
s
servative with respect (S,B,p) in the ergodic theory
roup
all
conservative if
con-
S (which are assumed closed mappings) are
It is shown that the
sense.
sem
is a left group. An extension of this result is
generated by the support of
obtained for o-finite p.
KEY WORDS AND PHRASES.
Topological metric semigroups, left groups, conservative
translations of the semigroup, Borel measure, r*-invariant measures.
1980
SUBJECT CLASSIFICATION CODE.
AMS
43A05, 28D05, 22A15
1. INTRODUCTION.
Throughout we shall be dealing with the measure space
a topological metric semigroup, B
U
ably additive Borel measure on B U.
s
sx,
{s
F
x
S,
B
Bx
S,
-1
={s;
F
sx
Bo,
V)
where
S
c
-
is
a non-negative count-
We shall assume that the right translations
S; every neighborhood of s has positive
is finite on compacta and
that
For
p
S (which are measurable) are closed mappings. The
x
defined by
(S,
and
its Borel G-algebra
support
F
tx:
of
measure). We assume
(=the complement of F)’is of p- measure zero.
B}.
The closure of B will be denoted by
The purpose of this note is to study the effects on S of certain invariant con-
ditions on the
tx’S
type especially that of conserva-
measure-preserving
of the
tiveness and r*-invariance. By abuse of language (interchanging the roles of the
and p) we shall say that
conservative
is
conservative with respect to the measure space
if every
recurrent)
(or
ix,
x
tx’S
S, is
in the ergodic theory sense
(S,Bu,p)
that is, if
Bx -I
:
for all 1
B
(implies)
r_
B
(Bx
-I
B)
(i.I)
0
x
Such transformations are familiar in ergodic theory (See
acterlzed by the property of (infinite) recurrence, i.e.,
$I
t./
V B BU x S,
{ B
m=l
[1]
Bx
and
-i
[2J
and are char-
0
(1.2)
Equivalent definitions of conservativeness" would be obtained using closed sets
in
(1.1) or requiring the non-existence of a closed set K of positive measure such that
-I
-2
Kx
x
S, is a disjoint sequence. The measure
K, Kx
N.A. TSERPES
196
if
invariant
Bo
B
For all
S,
x
(Bx
-1
(B)
(1.3)
Conditions (1.1) and (1.3), despite their similarity, are really independent. However
,
for bounded
trivially
r*-invariance
implies
This fact and the
conservativeness.
[3}
wish to generalize the followlng result of
(Theorem 1, below) lead us to consider
the o-finite case and the effects of conservativeness of p on S.
9..
THE CASE OF FINITE MEASURE.
[3]
THEOREM 1. (of
).
Suppose in
(S,Bu,p),
r*-invariant (not necessarily fi-
is
nite) measure
and the t x
x
S, are closed mappings.
and
F is left slmple
right cancellatlve
,
A simple generalization of this theorem (with a new proof) in the case of bounded
is the following
Suppose in (S, B ,p)
THEOREM 2.
always closed). Let D
t$F n
and let L
D
n=l
2
Suppose now that
u2(Bx-1
we see that
of x. Then
/
xS
the support
.
pk
the
k th
convolutlon power of p. For k
f (y- IB)
i.e.,
2,
(2.1)
(dy)
.
j > 2. Hence
U
VxJ-lx-
F @
D
is left slmple.
-
U Dx F @
and
It follows, since x was arbitrary, that
Also
IY
Dx
V
D
Sx
and a fortlorl
Dx
is left slmple semIEroup
and
L
L’
follows that
aa
-1
F
3
Case 2: Suppose
or
a+l
v E
a
and
D
Let
a
D. Since D
is
Then it
(Proc. Jap.
,p. 318), and being a compact semlgroup, it
Interlor(aa
> O. This tpltes by (1.2) applied to
It follows that there is
and
Incldentally, also
We consider two cases. Case 1:
-1
is compact by a result o Worlta and Hanal
Acad. 32 (1956), p. 10-14) (cf. also
contains an Idempotent.
S
for all x
Interlor(aa -1)
Frontlet(am
V
for all x
is left slmple since it is the mlnlmal left Ideal of S
a
,
U((y-lB)x
U(y-l(Bx -1- B)) u(dy)
(2.2)
0
y B)u(dy)
all powers k are also conservative. Let x
D and U any neighborhood
k(u) > 0 for some k, since D is the semlgToup generated by the powers of
-j
for some J
F of
By (1.2) applled to k we must have D/ Ux
t
left slmple
vaa J-1 a
F of
and both D and L are left groups
L
D
Then
Next, we show that there is an Idempotent element in D.
k(aa-1)
are assumed
t’s
x
B)
D, so that
hence Vx
L =/ Sx
xS
xES
Sx
(sy-l) (dy)
Bx-l B. Since
and we may assume
L’= C
is bounded conservative. (The
,(B)
(B)
Sx
p
be the subsemigroup generated by the support
PROOF. Let us denote by
U
is a left group, i.e.
F
Then
(aa-1)a-J/ aa- 1
k
-1)
that
for some
and hence the powers of a
F
Then for some k,
k{aa-l-- 0 (aa-1)a-i))=
J, so that
vaJa
a
0
or
form a finite semlgroup which
must contain an Idempotent element. Therefore, D is a left group, since an alternate
characterization of left group is that it be left slmple and contain an Idempotent
element.
3.
THE CASE OF INFINITE MEASURE.
One may wonder what is an appropriate condition under which an infinite
197
CONSERVATIVE MEASURES ON SEMIGROUPS
becomes conservative. One such condition (admittedly not very manageable)
variant
is the Don-existence of an unbounded (in measure)
some x
and (Gx
-1
G
set G such that
Gx
-1
G
for
Gx-l=)
G
sets G such that
G) > O, that is, for all unbounded
-1
O. Such a condition plus r*-invariance imply (1.1).
G)
G
for some x, we have (Gx
the
In the infinite case we have obtained only partial results summarized in
following theorem.
THEOREM 3.
tx,
translations
E
k=l
F
(ii)
If
(iii)
If
and
F (_ E
D
is u-finite and S is separable, then
S
F
E being a closed subsemigroup.
E =D
is the non-contractiveness of p
REMARK. A condition that makes F a subsemigroup
i.e., for closed sets B,
an
p(B) > 0
i+l
Hence
U
and
x E E. Using (1.9-), since
(ii):
(Bx
-1)
i-I
/ F
F
i+l
i.>k
Ux-i(
F
implies
is in E (E being an ideal in D),
p(x-lB),
[7]
and
U
B
E
B
F
U’
and hence
U
Let now
F
Ff i
be as above
Fxi-lx
U
we have
(we may take
xIEE
Ex
imply that the functions
are measurable and also the validity of (9..1) and (9..2)
). Hence, that the convolution powers of p are again conservative.
Then, as in Theorem 9., D is a left group. (the conservativeness of
left simplicity
f E F there is
of a point
for every k > 1.
The separability of S and -finiteness of
and
[6]
(cf.
Fx
Uf-i
U, such that by (1. 9-)
and
i > 2) and
p(Bx) > O, x E F.
implies
Let k be given For any neighborhood
i > k, depending on
t’ F
a left group.
F is a subsemigroup, then F is a left group.
or
PROOF. (i):
i__l
Then,
Sx}
Ex (/
xS
C /1
xE
(i)
x
F i+l
i_>k
.
is an (infinite) conservative measure and the right
F i and
Let D
S, are closed. Let F denote the support of
(S,B,)
Suppose in
and
pk
is needed for
to produce an ldempotent element in D,as in Theorem 9. ).Observe
that E being an ideal in D must equal to D since D is now left simple.
(_ili): We can prove F to be left simple
Fxxx-1
Fx
Fxx
-1
-1
and hence
p(Fxx
-1
either by observing that
Fx)
0
n
F,
Fx)
0
for x
by (1.1), so that B(F
Fx, being open, is empty, o., we may use (1.2) and the
To produce an idempotent we use
Fx
argument n Theorem 2 showing that F
k
-1
with k
1, as in Theorem 2.
the argument on the Interior(aa ), a E F, and
(.Here we don t need the convolution powers pk for k > 1 to be conservative).
sfnce
F
Fxx
and
F
REFERENCES
l.
2.
3
HALMOS, P.R. Lectures in Ergodic Theory, Chelsea Publishing CO., 1956.
WRIGHT, F.B. The recurrence Theorem, Amer. Math. Monthly, 6_8 (1961), 9.47-248.
TSERPES, N.A. and MUKHERJEA, A. .esures de probablt r*-nvariantes sur un
semigroupe metrique, C.R. Acad. Sco Paris Ser. A, 268 (1969), 318.
!
.Compact semi.too.logica s_emfroUps and weakly
almost periodic functions, Lecture Notes in Math. No 42, Springer, 1967.
4.
BERGLUND, J.F. and HOFMANN, K.H.
5.
BOURNE, S.
Sur la Note de MM. Tserpes et MukherJea:Mesures de probabilite
invariantes sur un semigroupe metrque, C.R.
S_--Paris Ser. A 269
(_1969), 1143.
Acad.=
N.A. TSERPES
198
6.
MUKIRJEA, A.
7.
MUKhRJEA, A. On the equation P(B)
Math. Soc. 6 (1973), 224-230
The convolution equation p
p,q of Choquet-Deny and relatively
invariant measures on semigroups, Ann. Inst. Fourier 21 (1971), 87-97.
SP(Bx-1)p(dx)
for infinite P,
J.London