A NOTE ON CONSERVATIVE MEASURES ON SEMIGROUPS

Consider (S,B,μ) the measure space where S is a topological metric semigroup and μ a countably additive bounded Borel measure. Call μ conservative if all right translations tx:s→sx, x∈S (which are assumed closed mappings) are conservative with respect (S,B,μ) in the ergodic theory sense. It is shown that the semigroup generated by the support of μ is a left group. An extension of this result is obtained for σ-finite μ.


INTRODUCTION.
Throughout we shall be dealing with the measure space (S, Bo, V) where S is a topological metric semigroup, B U its Borel G-algebra and p a non-negative count- ably additive Borel measure on B U. We shall assume that the right translations tx: s sx, x S (which are measurable) are closed mappings.The support F of defined by F {s S; every neighborhood of s has positivemeasure).We assume that is finite on compacta and F c (=the complement of F)'is of p-measure zero.
For x S, B S, Bx -1 ={s; sx 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 of the measure-preserving type especially that of conserva- tiveness and r*-invariance.By abuse of language (interchanging the roles of the tx'S and p) we shall say that is conservative (or recurrent) if every ix, x S, is conservative with respect to the measure space (S,Bu,p) in the ergodic theory sense that is, if Bx -I : B

2) m=l
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 K, Kx -I Kx Conditions (1.1) and (1.3), despite their similarity, are really independent.However for bounded , trivially r*-invariance implies conservativeness.This fact and the wish to generalize the followlng result of [3} (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.
THEOREM 1. (of [3] ). Suppose in (S,Bu,p), is r*-invariant (not necessarily fi- nite) measure and the t x x S, are closed mappings.Then F is a left group, i.e.
F is left slmple and right cancellatlve A simple generalization of this theorem (with a new proof) in the case of bounded , is the following THEOREM Suppose now that Bx-l B. Since u2(Bx -1 B) U(y-l(Bx-1-B)) u(dy) U((y-lB)x y B)u(dy) we see that all powers k are also conservative.Let x D and U any neighborhood of x.Then k(u) > 0 for some k, since D is the semlgToup generated by the powers of the support F of .By (1.2) applled to k we must have D/ Ux -j t for some J and we may assume j > 2. Hence U VxJ-lx-F @ and U Dx F @ and a fortlorl U Sx . It follows, since x was arbitrary, that IY Dx V for all x V and hence Vx D, so that D is left slmple.Also -Sx D for all x S and L =/ Sx is left slmple since it is the mlnlmal left Ideal of S Incldentally, also xS Acad.32 (1956), p. 10-14) (cf.also 3 ,p. 318),and being a compact semlgroup, it contains an Idempotent.Case 2: Suppose Interlor(aa-1) F Then for some k, k(aa-1) > O.This tpltes by (1.2) applied to k that k{aa-l--0 (aa-1)a-i))= 0 1 It follows that there is v E (aa-1)a-J/ aa-for some J, so that vaJa a or J-1 a+l vaa a a or a and hence the powers of a 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 CONSERVATIVE MEASURES ON SEMIGROUPS 197 variant becomes conservative.One such condition (admittedly not very manageable) -1 G for is the Don-existence of an unbounded (in measure) G set G such that Gx some x and (Gx -1 G) > O, that is, for all unbounded G sets G such that Gx-l=) G -1 for some x, we have (Gx G) O. Such a condition plus r*-invariance imply (1.1).
In the infinite case we have obtained only partial results summarized in the following theorem.(iii) If F S or F is a subsemigroup, then F is a left group.
REMARK.A condition that makes F a subsemigroup is the non-contractiveness of p i.e., for closed sets B, p(B) > 0 implies p(Bx) > O, x E F.
PROOF. (i): Let k be given For any neighborhood U of a point f E F there is an i > k, depending on U, such that by (1. 9-) Uf-i F and hence U' Ff i and U t' F i+l Hence F / F i+l i.>k for every k > 1.Let now U be as above and x E E. Using (1.9-), since Ux-i( F implies U Fxi-lx (we may take i > 2) and Fx i-I is in E (E being an ideal in D), we have F xIEE Ex (ii): The separability of S and -finiteness of imply that the functions (Bx-1) and p(x-lB), B E B are measurable and also the validity of (9..1) and (9..2) (cf.[6] and [7] ).Hence, that the convolution powers of p are again conservative.Then, as in Theorem 9., D is a left group. (the conservativeness of pk is needed for left simplicity and 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 either by observing that for x n F,

Fx
Fxxx-1 Fxx -1 and hence p(Fxx -1 Fx) 0 by (1.1), so that B(F Fx) 0 sfnce F Fxx -1 and F Fx, being open, is empty, o., we may use (1.2) and the argument n Theorem 2 showing that F Fx To produce an idempotent we use -1 k the argument on the Interior(aa ), a E F, and with k 1, as in Theorem 2.
(.Here we don t need the convolution powers pk for k > 1 to be conservative).
all 1 r_ B x Such transformations are familiar in ergodic theory (See [1] and [2J and are char- acterlzed by the property of (infinite) recurrence, i.e.,

L
show that there is an Idempotent element in D. Let a D. a result o Worlta and Hanal (Proc.Jap.

THEOREM 3 .
Suppose in(S,B,)    is an (infinite) conservative measure and the right translations tx, x S, are closed.Let F denote the support of .Let D i__l F i and 1 Ex (/ Sx} and F (_ E D E being a closed subsemigroup.xExS (ii)If is u-finite and S is separable, then E =D a left group.
2. Suppose in (S, B ,p) p is bounded conservative.(The t's are assumed x always closed).Let D be the subsemigroup generated by the support F of , i.e., PROOF.Let us denote by pk the k th convolutlon power of p.For k 2, 2 (sy-l)