A NOTE ON QUASI R *-INVARIANT MEASURES ON SEMIGROUPS

A characterization of quasi r*-invariant measures on metric topological semigroups is obtained by showing that their support has a left group structure thus generalizing previously known results for relatively r*-invariant measures and the topo-algebraic structure of their support.


INTRODUCTION.
Two interesting characterizations of absolute continuity of a Ra- don measure W with respect to a right Haar measure on a locally compact group S is the continuity of x+(Kx) for all compact K and also the con- dition .6 << for all xeS, 6 the point-mass at x, (1.1) x x (of.Ill,V, 19. 18, 19.27, 20.28, 20.3)).
One is tempted to replace k with U in (1.1) to obtain thus a measure U io n ou o e o ouex oontino (o. []a [3]).Moe- over this condition i.e., U x << for all xeS (1.2) makes sense even when there is no invariant and S is only a semigroup.
Following [2] and [3] we define a non-negative, inner regular, locally finite Radon measure on a semitopological semigroup S to be quasi r*-in- variant if in addition to (1.2), the companion condition <<* for all x x, also holds.Similarly one defines quasi l*-invariant measures using << 6 x* for all xeS and its "companion" condition i.e. the "nuiva- lence of these measures, where 6 .U(B)=U(x-B)=U{s; xseB}.In [h]A.Mukhel- X jei considered relatively r*-invariant measures satisfying U(Bx-I)=B(x)u(B) for all Borel B and x, with 8 being a fixed homomorphism 8 S (0,), and proved that their support is a left group i.e., left simple and right cancellative semigroup.Such measures are clearly quasi r*-invariant but n the non locally compact case there may exist quasi r*-invariant meas- ures although there is no right invariant one (cf.e.g., [5]).In this note we prove similar results (under weaker conditions) for quasi r*-invari- ant measures and study the topo-algebraic structure of their support F={xeS; every open neighborhood of x has positive U measure}.
Throughout we will assume that S is (at least) a T 2 semitopologieal semigroup in which the right (continuous) translations t :s/sx are closed x maps.Topologically, in most cases, we will assume (and state it explicit- ly when we shall do so) that S is either a locally compact paracompact countable or a metrizable space.The measure V is called purely atomic if every singleton in F has positive measure.Clearly, a purely atomic measure on a left simple support (i.e., Fx F for all xeF) is trivially quasi r*-invariant.Also, pure atomicity for o-compact S, forces F to be countable.Although the purely atomic case with the extra condition that -I V(xx )<, may have some interesting pathology, in this note however we consider mostly the non-purely atomic case and the continuous case (i.e. V(x}=O for all xeF) and show in this case that F is a left group and a right ideal of S. (See Theorem 4). 2.

STRUCTURE OF THE SUPPORT
The effect of the existence of an atom (singleton) on the structure of F is shown by the following.
THEOREM I. Suppose quasi r*-invariant on S and that the right translations t xeS are closed X' (i) If S is o-compact and if for some aeF, {a}>O, then aF s a countable left simple, left cancellable (separable metric) semigroup.
i) If for some beF, W{b}=O, then F is a left group.
REMARK.In this Theorem S is either a locally compact paracompact countable (or a metrizable) topological semigroup and the attributes in parenthesis hold if S is metrizable.
PROOF.First we observe that the condition <<-6 x for all x, imme- diately implies that FC_x closure (Fx); for V open in F implies D(Vx-1)>O which implies VFx # @.Also the companion condition ,6x<<W implies that F is a semigroup, in fact a right ideal For if (B)>O, then since BCBxx -I we have M(Bxx-1)>O and hence M(Bx)>0.(Of course, we need the semitopo- logical property of S).Hence, we have Fx F for all x and since the t lhh, Lemma 2.1.),that for every aeF, aF is left cancellable.Also aF is left simple, since F itself is so.(i): Since in this part we assume F o-compact, aF is also o-compact and all its elements have positive mea- sure, so it must be countable on aF is purely atomic (ii) Since {bb-1)=O, Interior (bb-1)= and so Frontier (bb -I) must be countably compact ([7], p. 25h, Exerc. 14)and by paracompactness, it is a compact subsemigroup and hence contains an idempotent element.It follows that F is a left group. (Observe that bb-1# since F is left simple).The rest follow easily.
REMARK.We remark that we could have assumed S to be only semitopo- logical (multiplication separably continuous) and again F ould turn out to be a topological left group in (ii) and aF a topological group by a theorem of Ellis (cf.[8], p. 60).Of cource we would have to impose com- pleteness in the metric case.(cf.e.g.N. Bourbaki, General Topology, Part II, p. 258).
We shall need the following tope-algebraic result which is also of independent interest.LEMMA I. Suppose S is a locally compact or complete separable metriz- able semitopological semigroup with the t (right translations) closed.
x Suppose further that F is a closed subset of S such that Fx xF F for all xeF.Then F is a topological subgroup.

PROOF.
Clearly, F-closure(FF) F and hence F is a subsemigroup.
Since xF=F for all xeF, it follows that Fa=F is right cancellable (-ee proof of Theorem I), so F is a left group.Since every aF contains now some eF, with ee e, and since eF is closed, we have aF=F for all aF, that F is also right simple.Hence, F is a group and being locally com- pact (or complete separable metric) is a topological group.THEOREM 2.
Suppose S is a topologically complete topological se migroup with the t closed and W is a quasi r*-invariant and also quasi x ,l*-invariant measure on F. Then F ia a locally compact group and is ab- solutely continuous with respect to the right Haar measure on F.

PROOF.
From the above Lemma we obtain easily F to be a Baire topological group.By the main result in [3], F is locally compact.Then it follows from the work of ([2], p. 229) that is the indefinite integral with respect to the right Haar measure of an almost everywhere (Haar) po- sitive function.
REMARK.Actually half of the definition of l*-invariance is requir- ed in the above Theorem, i.e., we only need <<x.. W for all x.Of course we need (ful.]y)quasi r*-invariance.
For continuous measures we have the following theorem.
THEOREM 3. Suppose S is a topologically complete semigroup with the t "s closed and W is a quasi r*-invariant continuous (i.e., W{x) 0 for x all xeF) measure on S. Then F is a locally compact left group.PROOF.Since the t are closed, the Proof of Theorem (ii) shows x that F is a left group.Now every left group satisfies condition (R) of [9] i.e., that K-I(Kx) is compact for compact K. Also since F is a Baire space, every point of F is a point of condensation (each neighborhood don- tains uncountably many points, since F has no isolated points by the "conti- nuity" of .).Now the proof given in ([9], Lemma I, p. 255) of producing a relatively compact neighborhood V of a point x, goes through since the process of producing disjoint translates of K (K of positive measure) by points of V (V a Baire space itself) must terminate after a countable e- numeration (local finiteness here is needed), and this leads to contra- diction.
Question: Can we say anything as regards to absolute continuity of such with respect to some "standard" r*-invariant product measure on % the right Haar measure on the group corn- the left group of the form W ponent (factor) of F?).
The following Theorem summarizes the various conditions on (i.e, "fibers" must not be "too big" in measure) and on the t in order that X the support of a quasi r*-invariant measure to be a left group.
THEOREM h.Suppose is a quasi r*-invariant measure on S (always the t are assumed closed) and further suppose that any one of the fol- x lwing three conditions obtains (a) is not purely atomic (this ho3ds in particular if U is continu- OUS) (b) The t "s are proper maps i.e., in addition to closedness we also X have yx compact for all x,y EF.
Then F is a left group.
Assume (b).Then aa is a compact semigroup (non-empty by left simplicity) and hence contains an idempotent element.Next, assume (c).Trivially one checks that 8 is a constant (equal to unity) on each -I -I a Now take zeaa Defining the support of a closed set in the natu- --I ral way, it follows that Support (aa 1)=aa -I and that aa is also left -I simple.For ueaa consider L (ZU)(ZU)-1 z(aa- -I with is non-empty by left simplicity of z(aa and every element in it has measure greater or equal to the measure of z, since zezuu and 8 is -I on aa It turns out that L must be a finite simigroup and hence must contain an idempotent.The part using Condition (a) follows from Theorem I. (of. the Remark to that Theorem).
We conjecture that one can dispense with the rather unnatural condi- tion (c).Observe that in (b) and (c) we only need S to be T 2 semitopo- gical (with the tx'S closed).

X
are closed we have Fx F. It follows by a standard argument in ([6], p.