Invariant Means on an Ideal

transactions of the american mathematical society Volume 288. Number 1, March 1985 INVARIANTMEANS ON AN IDEAL BY MICHEL TALAGRAND Abstract. Let G be a compact abelian group and Q an invariant ideal of I.X(G). Let Mq be the set of invariant means v on LX(G) that are zero on Q, that is p(Xa ) = 1 f°r Xa e Q We show that Mq is very large in the sense that a nonempty Gs subset of Mq must contain a copy of /?N. Lei Eq be the set of extreme points of Mq. We show that its closure is very small in the sense that it contains no nonempty Gs of Mq. We also show that Eq is topologically very irregular in the sense that it contains no nonempty Gs of its closure. The proofs are based on delicate constructions which rely on combinatorial type properties of abelian groups. Assume now that G is locally compact, noncompact, nondiscrete and countable at infinity. Let M be the set of invariant means on LX(G) and Mr the set of topologically invariant means. We show that M, is very small in M. More precisely, each nonempty Gs subset of M contains a v such that v(f) = 1 for some/ £ C(G) with 0 ^ / «; 1 and the support of / has a finite measure. Under continuum hypothesis, we also show that there exists points in Mr which are extremal in M (but, in general, Mt is not a face of M, that is, not all the extreme points of M, are extremal in M). 1. Results. Let G be a locally compact group. A left invariant Haar measure of G is denoted by dx. Whenever G is compact, we assume the Haar measure to be normalised. The measure of a measurable set A is denoted by \A\. For/g L°° = LX(G) and u G G, we consider the left translate/„ g Lx(G) given by fu(t) = f(ut). A (left) invariant mean v on G is a positive linear functional on Lx with v(l) = I and v(fu) = v(f) for / g Lx and u g G. We say that G is amenable when there exists a left invariant mean on G. We say that G is amenable when there exists a left invariant mean on G. We say that G is amenable as a discrete group when G, provided with the discrete topology, is amenable, that is, there is a left invariant mean on /°°(G). We denote by M the set of invariant means on G. We say that an invariant mean v is topologically invariant if, for /, <i>G Lx, <f>> 0 and f<p = 1, we have v(<p * f) = v(f), where <¡>*f(x) = ff(s~1x)<b(s)ds. We denote by M, the set of topologically invariant means on G. It is known that Mt # 0 whenever G is amenable. (For a proof, as well as for the proof of all these basic facts, see [7].) When G is compact, M, = {dx}. When G is not compact, and amenable as discrete, various known results (like [10, Theorem 6D]) show that Mt is very small in Received by the editors April 17, 1984 1980 Mathematics Subject Classification. Primary 43A07; Secondary 46A55. Key words and phrases. Invariant mean, invariant ideal, extreme point, exposed point, geometry of the set of invariant means. ©1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page 257 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 258 MICHEL TALAGRAND M. Our first result is a further step in this direction. We provide M with the topology induced by o(Lx*, Lx). Theorem 1. Assume that G is metrizable, countable at infinity, noncompact, nondiscrete, and amenable as discrete. Then, whenever H is a nonempty Gs of M, there exist v in M and a continuous function f on G, with 0 </< 1, v(f) = 1 and \{x;f(x)> 0}\ < +00. It should be noted that for A c G, \A\ < oo; then v(A) = 0 for v g M,. So the above theorem asserts that H contains a v which fails to belong to Mt in a spectacular way. When G is compact, the Haar measure is an extreme point of M. So when G is not compact, a natural question is to investigate the position of M, inside M. In a previous work, we showed that for G — R, there exist extreme points of Mt that are not extreme in M. Here we shall show that, surprisingly enough, some extreme points of M, can be extreme in M (or, equivalently, some extreme points of M can belong to Mt). Theorem 2. Assume continuum hypothesis (CH): If G is countable at infinity, metrizable and amenable, there exist extreme points of M which are topologically invariant. In fact, any Gs set Y of Mt that contains an extreme point of Mt contains an extreme point of M. We now turn to a different topic. Given a left invariant ideal Q of Lx, we say that the invariant mean v is zero on Q if v(A) = 0 whenever \A g Q. We shall study the set Mq of invariant means which are zero on Q. When Q is large (say maximal), Mq is much smaller than M. There are three fields of study: Case 1. Study of MQ for G compact. Case 2. Study of MQ for G noncompact. Case 3. Study of M, n MQ for G noncompact. We shall limit ourselves to the first case. The same results hold in the other two cases, and the ideas of the proofs carry over. It is of some interest to state the problem in another language. Using the Stone representation theory, we know that Lx = C(S), where 5 is the spectrum of Lx. The left action of G on Lx induces an action of G on S. An invariant ideal Q of Lx corresponds to an invariant closed set Q of 5", and MQ identifies with the set of probability measures on Q that are invariant under the action of G. Hence the nature of MQ is strongly related to the nature of the action of G on Q. In a previous work [11], we have shown that when G is nondiscrete and amenable as discrete, MQ contains a copy of ßN. The results we shall prove here are much more precise; they rely on some precise constructions of measurable sets; these constructions use the structure of G, and we have been able to perform them only when G is abelian. We do not have enough knowledge of nonabelian groups to be able to decide with reasonable effort whether the ideas can be adapted to the general amenable case and whether the results still hold in that case. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 259 INVARIANT MEANS ON AN IDEAL Theorem 3. Assume that G is compact abelian metrizable nondiscrete. Then for any proper ideal Q of Lx, every nonempty Gs of Mq contains a norm discrete copy of ßN. It follows in particular that MQ has no exposed point. This answers a question of E. Granirer [6]. We denote by EQ the set of extreme points of MQ. Theorem 4. Assume that G and Q are as in Theorem 3. Then Eq contains no nonempty Gs of Mq. Theorem 5. Under the same hypothesis, EQ contains no nonempty Gs of Eq (and hence is very irregular topologically). We assume in all these results that G is metrizable. This restriction is inessential and can be lifted by standard techniques. The author is indebted to Professor E. Granirer for inviting him to the University of British Columbia and arousing his interest in this field. 2. Proof of Theorem / g L°°, we set 1. Given a sequence m(u,x)(f) u = (ux,...,un) of G, x g G and = n'1 £/(«,■*). i^n Given x, this quantity is not well defined. However, changing / on a negligible set changes m(u, •)(/) on a negligible set, so the conditions we shall write depend only on the class off. Given two sequences u = («,,.. m(u- .,«„), v = (vx,... ,vm) of G, and x g G, we write v,f)(x) = (nm)~ £ f(uiuJx). Z< n ,j^m We say that a set IF c Lx* is elementary if it is of the type W = {m g Lx*;Vi where/, < n,m(fl) g [a,.,è,]}, g Lx, a¡, b¡ g R. The following lemma is standard. Lemma 1. Let W be as above. Assume that G is amenable as discrete and that, for each finite sequence u of G, there is a finite sequence v such that \{x g G,;Vi < n,m(u ■v, x)(f) g [a,, bi]}\> 0. Then M n W * 0. We now prove Theorem 1. First step. Let N= {p,€EM;3geC(G);0<g<l, \{x; g(x) > 0}\< oo;p(g) = l}. For p g M and h g Lx, we notice that ¡i(h) = 0 whenever h has a compact support. Indeed, for each n, there exist u,,... ,un G G such that ||EI<fl hu || = ||/z||, so n\n(h)\ s$ ||A|| and fi(h) = 0. So, for p g N, for each compact set K of G, and for each r/ > 0, there is g' g C(G), 0 < g' s£ 1, \{x; g'(x) > 0}\ < r/, g' = 0 on K, and License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 260 MICHEL TALAGRAND H(g') = 1. Indeed, if 0 < g < 1, g g C(G), p(g) = 1 and \{x; g(x) > 0}| < oo, it suffices to take g' = hg, where 0 < h < 1, h g C(G) and h = 0 on a large enough compact set. Let us show that if (p„) is a sequence in N, its closure is contained in N. Let Kn be a sequence of compact sets, with Kn c Kn+X and G = UAT„.There exists a sequence (g„) of C(G), with 0 < g„ < 1, g„ = 0 on K„, p„(g„) = 1 and \{x; g„(x) > 0}\ < 2"". Let g = sup„g„. Then g g C(G), 0 < g < 1 and |{jc; g(x) > 0}| < oo. Since, for each n, p„(g) = 1, we have p(g) = 1 for each cluster point p of (p„), which proves the claim. Second step. We show that N is dense in M. Since N is convex, it is enough to show that, given/in Lx and a in R, IrneJIi, w(/)>a=>3pGA, ¡i(f) > a. We can assume that 0 < / < 1. For rVinteger, let/* g L°° given by/A(x) x g ATA., and/A(x) = -1 for* <£ AT*.We have w(/*) = w(/) > a. = f(x) for Given u = (ux,.. .,un) G G", let C,k,n xG Gízz"1 £/*(«z*) (= 1 > a . Since m(n~1I."=lfuk) > a, we have |C,(A"|> 0. It follows from [10, Lemma 6C] that there exist open sets ß*-" of G with |Í2A"| < 2" such that bk-"-"(t, u) = C*-"n fl z,ß*'"> 0, Vz7g N,Vz = (r,,...,^) g Gq. It is routine to check that the map (t, u) -» bkn,q(t, u) is lower semicontinuous. particular, its infimum on a compact set is > 0. Since Inf{|Cuu n ro1-1!; t,u^ there is a compact set L[ c ßu In Kx) > 0, such that Vr, m g Kx, |C„u n rL,| > 0. Let sx be large enough that Ll c Ks _, and r2 large enough that K2K2KS c ATr. In the same manner, there exists a compact L2 c W2-2such that Vi*,,^) g a:22,Vh *f, c;-2 n txL2 n z-2L2|> 0. 1 on K2KS¡. This shows that C^-2 n ÄT^ = For z; G AT2,we have/(,r2(x) for w g A"2. So we can assume L2C\ Ks „, = 0 by replacing L2 by L2 \ KSi. In this manner, we construct compact sets Lp, integers s and rp with Lpc Ks _x, Ln c ßVandL„+1 n AT,_, = 0,and a:;,vm vr=(r1,...,ij *;. Q"' n fl ttLp> 0. ¡ap We have |Z.^| < 2~r» < 2'p so there is g g Cíe) with g = 1 on L = UpLp and |{ g > 0} | < oo. Moreover, asf^f x g G; p"1 £/„(*) for each integerp, ** a; V/ ^/>>* e r/L > 0, Vi*,,...,!*., *,,...,*. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use G, 261 INVARIANT MEANS ON AN IDEAL sO also e G] p~l L /«,(*) > a, Vz < p, u,x G L i<p > 0. So Lemma 1 shows that there is v g M with v(f) > a and v(L) = 1, so v(g) = 1. 77zz><7 Step. Let zn g 77 and let (F„) be a decreasing sequence of closed neighbourhoods of m with C\V„c 77. Since N is dense it meets each V„, and the first step shows that N n (¡„Vn =* 0. The proof is complete. 3. Proof of Theorem 2. Let B he the unit ball of Lx, provided with the weak »-topology. It is a compact metrizable convex set. A Borel probability X on B has a barycenter bx in B, given by bx(g) = fBf(g) dX(f) for g g Lx(G). We consider a p g L°°* as a function on B and we set, accordingly, essinfA/x = Sup{Inf{p(/);/G^l};^ Definition Borel, X(A) = 1}. 1. We say that p is submedial if for each probability À on B we have essinfAp < p(¿>a)- Theorem 6. Assume that p is submedial and an extreme point of Mr Then p is an extreme point of M. Proof. According to [4] it is enough to show that, for/ g Lx, 0 < / < 1, we have Inf,fi(ff) < n2(f). Let K he a compact set G of positive measure and rj the normalisation of the restriction of the Haar measure to K. Fix v g G and let <f>: AT2-> B be given by <p(t, u) = fju. For g g Ll(G), the map (t,u) ^ j f„,(x)fu(x)g(x) dx is continuous. It follows that <j>is continuous. The image measure X of tj X 17by <}> is supported by <j>( K X AT).Since pis submedial, there is?, u g K with p (/„,/„) < p(^). An easy computation shows that ¿> = /Vz,,,where A(w)=|A:r1//('w)i//. So Infp(//) f = lnf fi(flvfu) « Infp(/z/z„). z,t>,« u Since p is extremal in Mt, the Proposition 4A of [10] shows that Inf,,p(/i/z,,) < p2(/z). Since fi is topologically invariant, p(A) = p(/), which finishes the proof. Theorem 7 (CH). 7/ G z's countable at infinity, amenable, and metrizable, each Gs set Y of Mt which contains an extreme point of M, contains an extreme point which is submedial. Proof. Let m be an extreme point of Mt contained in H. Let Vnbe a sequence of neighbourhoods of m with f)Vn c 77. Since Mt is a Choquet simplex, it follows from [1] that there is/„ g Lx, ||/„|| < 1 with m(fn) = Sup{p(/„); p g M,) and Vp g M„ M/B) = "»(/„) => p G F,,. Let /= E2"/„. Then Vp g M„ p(/) = m</) ^ p g 7/ so we can assume that H = {p g A/r; p(/) = m(f)}. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 262 MICHEL TALAGRAND Since G is amenable, there is a sequence (Vn) of compact sets such that if, for x g G, we define mn(x) g Lx* by V/eL", «,.(*)(/)-|KJ~7 /(tt)ift, then each cluster point of a sequence w„(x„) belongs to M, [5]. If ß denotes the first uncountable ordinal, (CH) means that it has the power of R. So there is an enumeration (Xa, Aa)a<a of the couples (X, A), where À is a probability on B, and A is a Borel set with X(A) = 1. Recall that a subset F of a convex set L is called a face whenever F is convex, and Vjc, y G L, (x + _v)/2 G F=> x,_y G F. By induction over a, we construct a decreasing sequence (Ha)a<a of closed faces of Af, such that for a > 1 the following conditions hold: (a) 77a is a Gs. (b) 3/a g Aa, Vp g 77a, p(/J < u(bK). The induction starts with 770 = 77. Assume now that the construction has been done for each ß < a. Let Fa = f)a<aHß. It is a closed face that is a Gs. Let a = lnfliSFp(bxJ and F'a = {p. g Fa; p(z3A ) = a). This is closed face of Mt, that is a GÄ. So we can write F'a = Ç\V„, where Vn+ X c Fn and each Vn is of the type {p g Af,; p(hn) > an}. Let p g F,- Since p is topologically invariant, we have for each p that p(hn) = p(mp(-)(hn)) > «„, so there is tp g G with mp(tp)(hn) > a„. Since each cluster point of the sequence mp(tp) is in A/,, forp large enough, we have ™p(t)(h„) > a„ => mp(t)(hq) > aq Vq ^ n. So there is a sequence mn = mp (tn) such that mn(h ) > a for q < zz. Each cluster point p. of (mn) is in M, and satisfies p(h ) > a for each q, so is in F'a. In particular, a = hmmn(bx). Since m„ e L'(6) and Xa(Aa) = 1, we have m„(bK) = jAm„(f) dXa(f), so Fatou's lemma shows that /^ lim inf„/?!,,(/) dXa(f) < a. In particular, there is /« e Aa with hminf„ mn(fa) < a. This shows that 6 = Inf{p(/J; p g F^} < a. We now define Ha= {p ^ F'a; p(fa) = b). This finishes the construction. Now F = ria<ß 77a is a closed face of A/,, that is contained in 77, and condition (b) shows that each p g F is submedial. Since a closed face contains an extreme point [3], the proof is complete. The definition and name of submedial means are inspired by Mokobodzki's medial limits [8]. A natural definition is Definition 2. We say that p g Af is medial if, for each probability X on B, p is À measurable,and if Jp(f) dX(f) = p(JfdX(f)). The existence of invariant means which are medial follows easily from the existence (under (CH)) of a medial limit. However, it is worthwhile to note that medial means are never close to being extremal: Proposition 1. Assume that Z or R is a quotient of G. If p G A/f is X-measurable for each probability X on B, then p is not in the closure of the extreme points of Mt. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use invariant means on an IDEAL 263 Proof. Let 6 be a homomorphism of G onto 77, with 77 = R or Z. Write 77 = U7„, where 7„ is a half-open interval with |/„| > zz.For a subset A of N, let <t,(A)=n «-'(/„)• ne ,4 Let p be in the closure of the extreme points of A/,. Then the Theorem 2G of [10] implies that p(<p(A)) G {0,1} for each A. Let X he the Haar measure of {0,1}^. Since p ° (¡>is a zero-one additive map on P(N) with p ° <b(N) = 1 and p ° <p({n}) = 0 for each zz, we known that such a map is not A-measurable so p is not <i>(A)-measur- able. 4. Proof of Theorem 3. We recall an easy fact [2, paragraph 2, exercise 11]: Lemma 2. Given n real numbers xx,...,x„ in [0,1], ûh'î/ e g N, i/zere ex/sis a g N st/c/z f/iaf a < e" and that, for each i < n, there is /c, g Z with (i) From k - v*l < lAflnow on we consider additively. only abelian groups and denote their operation The proof of Theorem 3 will rely on the possibility of constructing sets of small measure, but that are big in other respects. The exact property needed is unfortunately complicated, but we shall single it out in order to avoid frequent repetition of its lengthy definition. Definition 3. We say that the compact abelian group G satisfies property (*) if, given e > 0 and p g TV,there exists q g N depending only on e and p, such that, given any number xx,...,x„ of elements of G, there exists a set A c G with \A\ < e, such that, given yx,...,y in G, the group G can be covered by at most q translates of thesetÇ)^pj^n(xJ+yi + A). We know no way to comment clearly on a property involving six quantifiers, but property (*) is much simpler than a first look might indicate. To understand it, the reader should analyze in detail the proof of Lemma 3, and make a picture of the sets involved. He will realise that the idea is elementary. Our first aim is to show that compact infinite abelian groups satisfy property (*). We first study some special cases. Lemma 3. T = R/Z satisfies (*). Proof. If we did not have to take xx,... ,xn into account, it would be enough to produce A such that, for each>»,,... ,y , n/5;/,(j, + A) contains an interval of length greater than some a > 0, and take q > 1/a. The idea is that Lemma 2 shows that any xx,. ..,xn g [0,1] are approximately of the type k¡/a for some a g N. So if one constructs a suitable subset of [0, a] and reproduces it by periodicity, we do not have to take xx,...,xn into account. However, some care is needed to control the perturbation created by the fact that x¡ not is exactly equal to kja. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 264 MICHEL TALAGRAND Let h G N with h 3= 2p/e. We shall show that one can take q = hp. Let xx,... ,xn g [0,1], let e = 2hp, and let a g N be as in Lemma 2. For 1 < z < p, we set B,= IJ o«z<// ' I/ah''1 +[0,2/aÄ']; 4 = U A:/*+ 5,.; ^ = U A,. Q*ik<a ;'<p We note that \B\ < 2/Aa and |^,| < 2//z, so |/1| < e. Also, \A\ = \A\. Now let yx,...,yp be elements of T. By induction over r < p, one checks easily that fli<r(.y; + A¡) contains a set of the type U0<jt<aA:/a + 7, where 7 is an interval of length 2/ahr. Now (1) and the choice of e, a show that f)i<p /<„(*/ + .V,+ ^) contains a set of the type U0ssA<a^/a + 7, where 7 is of length l/ahp. It follows that T can be covered by h p translates of this set, and this finishes the proof. Lemma 4. Assume that G is compact abelian, nondiscrete, totally disconnected, that the elements of G are not of uniformly bounded order. Then G satisfies (*). and Proof. The hypothesis implies that G has quotients that are cyclic groups of arbitrarily high order. The proof will be a "discrete version" of the proof of Lemma 3. Let b G N with 1/b < e/4, and q = bp. Let x,,... ,xn G G. There is a quotient 77 = Z/cZ of G, where c ^ 2b2p/e. Let Zj g [0, c - 1] be the image of xr According to Lemma 2, there is 0 < a < bp such that for/ < zzthere exists k¡ with (2) \zj/c - kj/a\ < l/aèp, i.e. \zf - k]c/a\ < c/abp. For z < p, define fi, c 77 = [0, c - 1] by x G Bt. » 3A: G N, kc/ab^1 0 < Â:< a/V-1, < jc < kc/ab*~l + 2c/ab'. One checks easily that |77,| < e/p. Let 5 = Uí<p B¡ and let v>... ,y'p be elements of 77. By induction type over r ^ p, one checks that 7)r = f]¡^r(B¡ + y¡) contains a set of the U {x: kc/a + ar *g x < ztc/a + <*,.+ 2c/a¿/} 0 < A< a for some ar g 77. Using this result for r = p, we see from (2) that D (zj + yf + B) /'</>,j^n contains a set of the type H {x; kc/a + a < x < A:c/a -I- a + c/a¿/'} 0< k< a for some a g 77, so 77 can be covered by q translates of this set. It suffices to take for A the inverse image of B in G. Lemma 5. Let G be abelian, compact, nondiscrete, totally disconnected, and such that each element of G is of order < b for some b. Then G satisfies (*). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 265 INVARIANT MEANS ON AN IDEAL Proof. We show that any q > (b/e)p will work. Let xx,.. .,x„in G. The subgroup F they generate is finite. Each finite quotient of G is a product of cyclic groups of order < b. So there is a finite quotient of G/F that is of the type Ylt< Gt, where each G, has a cardinal between e/p and be/p. Let p, be the canonical morphism from G onto G, and let ei be the unit of G,. One can take A = Vi¡^pA¡, where A,=p-l({e,}). Theorem 8. A compact infinite abelian group G has property (*). Proof. It is obvious that G has property (*) whenever one of its quotients has it. But either R/Z is a quotient of G, or G satisfies the hypothesis of either Lemma 4 or Lemma 5. The following is the main tool for the proof of Theorem 3. Proposition 2. Let G be abelian compact nondiscrete. Let Q be an invariant ideal, f G L°°, and a = sup{p(/):pG MQ). Let X a G with \X\ < 1/4. Let e > 0 and a G N. Then there exists B c G\X with \B\ < e, such that for each sequence w = (wx,...,wa) of G, there exists two sequences u = (ux,... ,uh), v = (vx,. ..,vc), there exists a measurable set F with G\F G Q, and there exists tj > 0 such that the set {x; m(u,x)(xFf) > a - r¡} can be covered by finitely many translates of the set {x g G; m(x ■v, x)(xFf) Proof. Let V = G\X. such that > a - e, m(w ■v, x)(xB) > 3/4}. Since Fis open with \V\ > 3/4, there exists t = (i,,.. .,td) (3) m(t,x)(V)>3/4 for each x in G. We know that G satisfies property (*). Let q = q(e/ad, ad). Let r/ = e/2q. The definition of a shows that there is a measurable set F with G \ F G Q, and z = (z,,... (4) Now property w = (wx,...,wu) ,z„) such that m(z, x)(xFf) < a + r/ a.e. (*) shows that there is A c G with \A\ < e such that, for each g G", there exists yx,...,yq g G such that G = (Jk<q(-yk + C), where C = Ç\(-Zj-t,-w, + A) (where the intersection is taken for/ < n, i < d, I < a). We set 7? = A \ X. Let us denote by v a sequence consisting of the points z + t¡ + w, for j ^ n, i ^ d, I ^ a, and let U he a sequence consisting of the points yk + z¡ + t¡ + w¡ for k < q, j < n, i < î/, / < a. Given x g G, there is k < z/ such that jc' = x + yk g C. It follows that m(v, x')(A) = 1. Since (3) implies that m(v, x')(X) < 1/4, we have m(v, x')(B) 3/4. Assume now that m(u, x)(xFf) > « - r/- We have a - r, ^ zn(M,x)(x/r/) = q~l L wi(i>,x +^)(Xf/)- License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use > 266 MICHEL TALAGRAND Since, for each s, (4) implies that m(v, x + ys)(xFf) ^ a + t], this forces m(v, x')(xFf) > a — 2öt) > a — e. The proposition is proved. We note that in the proof it is essential to be able to choose q independent of n, since zz depends on r\, and hence of q. This is what motivated the introduction of property (*). Proof of Theorem 3. First step. Let Y be a Gs of MQ, and m g Y. Let (F,) be a sequence of neighbourhoods of m with 77 = Ç)Vnc Y. Denote by Eq the set of extreme points of Mq. From Krern-Mil'man's theorem, there is, for each n, a kn g N, and pn _,,... ,pnk:< e Eq with E/asA/ia„ ,p„, g F„ for some «„., > 0, £/<* a/i./ = 1- Each pnj has a basis of neighbourhoods consisting of slices of MQ [3] so there exist hni g Lx and /3„, g R such that /v,(/z„,,) > ßn,„ and if *>,(/,„,,) > A,., for i < fc„, z>;g Mq, we get E,<A a,,^, g V„. We denote by (gJ) (resp. (ßs)) an enumeration of the hni (resp. /3„,) as a single sequence. Let as = sup{p(gs): PG Mq}. Second step. We construct disjoint open sets (Ain) of G for zz g N and i «S 22", with \A¡n\ < 2~2'~"~3, and such that whenever w = (wx,...,wn) G G" and s < zz, there exists two sequences u, v of points of G, r/ > 0, and F with G \ F g zO, such that the set (5) {x;m(u,x)(xFgs)> a2 --q) can be covered by finitely many translates of the set (6) [x g G; m(w ■v, x)(xFgs) > ots - n~x;m(w ■ v, x)(xA.J > 3/4}. The possibility of this construction follows from Proposition 2. Third step. For a g {0, 1}n, write as o\n the sequence of the first zzth terms of o. We identify [1,22"] with {0,1 }D»,where D„ = {0,1}". Fora g {0,1 }n, let Ua = \Jkn, where the union is taken forzz g N and all the k g {0,1}D" with k(a\n) = 1. Now let P, R be two finite disjoint sets of {0,1 }N. There exists zz0 such that, whenever a, p g P u R are distinct, we have a|zz0 =£ p|zz0. So, for zz > zz0, there exists kn g {0,1}D" which has value 1 on the elements a|zz for a g P, and value zero on the elements p|zz for p G R. Let w G G" and consider the sets K, L given by (5) and (6), where i = kn. By construction we can write K c U/<17y¡ + L. Let F' c F with G\ F' g G\ If i; = (z;,,.. .,u0), let F" =Cl(y,w, - vf + F') (where the intersection is taken over / < q, i < n, j < a). Since G\F" g g, the definition of as shows that K n F" has positive measure. It follows that L n (F" - ^7) has positive measure for some / < o. We note that for x g F" - V7,we have w, + i>.+ x G F for all i < zz,/ < a, so m(w ■v, x)(gs) {x G G; m(w ■ v, x)(gs) Va g F,m(w has positive measure. = m(w ■v, x)(xFgs)> as - zz"1; w(w In particular, • v, x)(F') - iz, x)(í7a) ^ 3/4; Vp g R,m(w = 1; ■v,x)(Up) < 1/4} So Lemma 1 shows that there is p g A7 with p(gs) > a\ p(F') = 1, p(Ua) ^ 3/4 for a g F, and p(Up) < 1/4 for p g R. By compacity, given any set P c {0,1}N and í g N, there is p g A/g with p(t/0) > 3/4 for a G P, p(Ua) < 1/4 for a £ F and p(gs) = ots. The first step and compacity License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use again show 267 INVARIANT MEANS ON AN IDEAL that there is p g 77 with p(Ua) > 3/4 for a g F, and p(Ua) < 1/4 for o £ P. The map p -» p(Ua) is continuous. Let <f,a(p) = Sup(lnf(p(/7a),3/4),l/4) and $: A/0 -» [1/4,3/4]<01) and <p(H) ^ {1/4,3/4}(0,1> lifted in 77, the result follows. be given by <t>(p)= (<ba(p)). This map is continuous . As this later set contains a copy of ßN that can be Remark. We have, in fact, constructed 2cardR disjoint decreasing intersections of slices and that all meet 77. sets of MQ that are 5. Proof of Theorem 4. It is known that M is very large (for example see [9]). The difficulty in proving Theorem 3 was that Mq can be much smaller than A7. The same difficulty will arise in proving Theorem 4. However, there seems to be another difficulty since we have not been able to find a simple proof of the much weaker fact that the set of extreme points of A7 is not dense in M. The method of proof will use the ideas of the previous paragraph, together with some new methods. The key will be a refinement of property (*) that we single out despite its complexity. Definition 4. We say that the compact abelian group G satisfies property (**) if, given e > 0, p g N, Fa neighbourhood of zero in G, and for each r < p a function <brfrom Gr to G, there exists q g N, depending only on the previous data, such that, given any number xv... ,xn of elements of G, there is a set A c G, a neighbourhood IF of zero, and functions \pr: Gr —>G with the following properties: (7) For yx,...,y in G, G can be covered by at most q translates of the set ^j<n,i<p(Xj+yi + A). (8) For u G Gr, write u = («,,... ,ur). Then \D\ < e, where D = \JW +■(«,- Uj) + ß*r(u) + A, the union being taken for r < p, u g G', z, j < r, and ß g {-1,1}. (9) Vr<p,V«GGr, Vv(m) + IFc <¡>r(u)+ V. Condition (9) means that \pr is close to <br. Given r, i, j < p, the union of W + ui — uf + A for u g Gr is all G. The use of \¡/r is to control the size of D. Condition (8) is fairly strong, and needs a very accurate choice of \pr to hold. It is hence surprising that the following should be true. Theorem 9. A compact infinite abelian group G has property (**). The plan of the proof is similar to that of Theorem 8: using the quotient, one reduces to the three cases considered in Lemmas 3-5. As the proof is fairly long, we shall treat only the case G = R/Z. The idea can be adapted to the other cases. First step. Let AeN with h > 6p/e. Let 1= {(i,j,r,ß);i,j^r^p,ß^ {-1,1}}. We enumerate 7 = (kdi^h- According to Lemma 2, there is a g N such that for /' < zzthere is k,. g N with (10) \xj-kj/a\^(10ahpb)'1. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 268 MICHEL TALAGRAND We show that q = hph works. We can assume that V = [-a1, a~l], since a can be taken arbitrarily large by decreasing e if necessary. Second step. For / < pb let c(l) = (ah')1. By induction over / < pb we construct maps <¡>'r from Gr to G, such that the following condition holds: (11) \/r*ip,Vu£Gr, ^(M)G^-1(M)+[-c(/),c(/)]. (12) If m = I - b[l/b] and if £m = (i, j, s, ß), for tz g Gs the number u, - u} + ß<b's(u) is of the type kc(l) for some k g Z. The induction step to / + 1 goes as follows: If m = I + 1 —b[(l + l)/b] and em = ('> 7> s> ß)i f°r r ^ s just let 4>l+l = <j>',while for r = s set <t>l;\u) = ßc(i + i)[c(i + îy'iu, - uj + ßu(u))] - ß(u, - uj). This completes the induction. We set \pr = <brphand W = [-c(pb), (9) follows from (11). For t < p, let c(pb)]. Condition AT= (J kc((T-l)b)+[-c(rb),c(rb)]. AeZ Each (z, /, r, ß) G 7 is of the type ¿m for some m ^ b. Fix t < p, and let / = w + t¿. It follows from (11) and (12) that 77= U {W+Ui-Uj-r »6C ß^r(u))c IJ kc(l)+[-3c(l+ AeZ l),3c(/+ 1)]. So \AT + H\ < 6/h; if A = \JT^pAT, condition (8) follows. The proof that condition (7) holds is just as in Lemma 3. Q.E.D. The proof of the following is identical to that of Proposition 2, using property (**) instead of property (*). Proposition 3. Let G be abelian compact nondiscrete. Let Q be an invariant ideal of Lx,fe Lx,and a = Sup{p(/);p g Mq). Let X a G with \X\ < 1/10. Let e > 0 and n g N. Let V be a neighbourhood of the identity, and for r < n let <prbe a map from G'to G. Then there exist a set B c G\X, a neighbourhood W of the identity, and for r < n maps \prfrom Gr to G, such that the following hold: (13) For each sequence w = (wx,...,wn) of G, there exist two sequences u = (ux,...,uh), v = (vx,...,vc), there exists a measurable set F, with G\F^ Q, and there exists r/ > 0 such that the set {x G G; m(u,x)(xFf) ^ a ~ v} can be covered by finitely many translates of the set (14) {x g G; m(w ■v, x)(xFf) > a - e; m(w ■v, x)(Xb) > 9/10}. Vz- < n, Vzv G Gr, ^r(zz) + IF c <br(u) + V. (15) |F| < e, where E = \JB + u,. - Uj + ß4>r(u) + W, the union being taken over all r < n, i, j < r, u G Gr, ß g {-1,1}. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use invariant means on an ideal 269 Proof of Theorem 4. First step. Let Y be a Gs subset of MQ and me Let IF„ be a sequence of convex neighbourhoods contains an extreme point, so it also contains Y r\ EQ. of m, with DIFn c Y. Each IFn a nonempty set of type {v g Me; '(&,) > ß„), forg„ 6 L», A, 6 R- Leta„ = suV{v(gn); v e MQ). Second step. We now construct two sequences (An), (Bn) of disjoint open sets of G, a sequence (F") of neighbourhoods of the identity and for r < n maps <}>" from Gr to G, such that the following properties are satisfied, where (k(n)) is a fixed sequence valued at any given integer infinitely many times: (16) Vr<z!,VizeG', 4>"+ l(u) + V" + 1 cz ^(u) + V, and V" is of diameter < 2~". (17) Let D„ = {jV" + ui-uJ + ßt?"r(u)+An, *,-UP"+ «,-«/ +/»#(«)+•*,,. where the union is taken over all choices of r < n, i, j < r, i #/, jSe {-1,1}, u g G'. Then |2>J < 2'7-B, |£J < 2"7"", Bm n Dn = 0 for m > zz,¿m n F„ = 0 for m > n. (18) For each zz, and each w g G", there is a measurable set F with G\ F g g, T) > 0 and two sequences «, z; of G such that when F is either ^4,, or Bn, the set (x g G; w(u, x)(xFg*{n)) > afc(B)- r,} can be covered by finitely many translates of the set [x g G; zn(w • v, x)(xFgk(n)) > o^j-n-1; m(w ■v,x)(xE) > V1"}- The first step is similar to the general step, so we assume the construction has been done up to n. Let Xx = U/<n E¡. Then \XX\ < 1/10, so Proposition 3 shows that there is a measurable set An+X c G\XX, a neighbourhood IF of the unit, and for r ^ zz maps \pr from Gr to G, such that the following hold: (19) For each sequence w = (wx,... ,wn) of G, there exist two sequences u, v of G, a measurable set F with G \ F g g, and r; > 0 such that the set [x g G; m(u, x)(xFgk(n + X)) > otk(n + X)- ij} can be covered by finitely many translates of the set {xG G;m(w- v,x)(xFgk(n (20) Vz- <zi,Vize + X)) > ak(n + X)- n~l; m(w ■ v, x)(xAk(i}+¡)) 3* 9/10). Gr, ^,(u) + IF c #'(«) + Vn + l. (2Y)\D\ < 2'1-", where ö - lM„+i+ «,- «7+ W,(«) + ^ the union being taken over all r < zz, z, / < z-, w g Gr, ß g {-1,1}. For m g Gn+1 we define ^b+1(ii) as the identity of G. Let X2 = 7) U U/0l />,. Then |^f2| < 1/10. Using Proposition 2 again, we find a measurable set 7f„+ 1 c G\X2, a neighbourhood K" + 1 of the unit, and, for r < zz + 1, maps <i>" + 1 form Gr License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 270 MICHEL TALAGRAND to G such that <¡>" + l(u) + V" + l c ¡pr(u) + W for r < n, u g G, such that |F„ + 1| < 2~7~", and that (18) holds for E = Bn+l. This completes the construction. We set <f>,.(u)= limn <t>';(u)for r G N, u G Gr, and ,4 = \JA„, B = \JB„. Third step. Using a method similar to that of the proof of Theorem 4, one sees that there exists Pi, p2 G f)„W„ with px(A) > 9/10, p2(B) > 9/10. Let p = (px + p2)/2. Thenp g C\nWn n Y. Let IF= (0G MQ;0(A) > \/4;9(B) > 1/4}. We show that W C\ EQ= 0 (so that p £ EQ). Suppose, if possible, that IF n EQ # 0. Then there is g g L00 and r/ with 17< a = sup{ v(g); v g A/e} and (22) Vz-gA^, r(g) > T)=> rG IF. The definition of a and Lemma 2, show that there is a measurable G\FX e Q, and a sequence « of G, such that for each sequence v of G m(u ■ v, x)(Fx) = 1 => m(u ■v, x)(g) < a +(a set F, with — i})/7. We can assume that u has at least 10 elements. Again according to Lemma 2, there is a measurable set F2 with G\F2 ra(w-z;,x)(F2) e Q, and = l,m(M-z;,x)(g)>r/ => m(u ■v, x)(A) > 1/4, m(u ■v, x)(B) > 1/4. By w>denote a sequence with w = u ■v. Let v G M^, with zj(g) = a. Then we have v(m(w, -)(g)) = Kg) = a. Since ?({*; ».(w, x)(F,) = 1}) = 1, we have v({x; So m(w, x)(g) < a +(a v({x;m(w,x)(g)>r1})> — r/)7}) = 1. 7/8. Since v({x; m(w, x)(F2) = 1}) = 1, we get v(Cx) > 7/8, where (23) Cx = {x; m(w,x)(A) > 1/4; m(w,x)(B) > 1/4}. Let n be the number of elements of w. Let A' = U(5¡„ A¡ and B' = U,^,, 5,-. We have |/F|, |7Ï'| < 2"7 from (17). We have v(m(w, -)(A'))= (24) p({x;m(w,x)(A')^ v(A') < |^|, so 1/8}) *i2-4, and a similar inequality for ZF. Let A" = A \ A', B" = B\ B', and let C2= {jcg G;m(w,x)(A") > 1/8; m(w, x)(7i") > 1/8}. It follows from (23) and (24) that v(C2) > 3/4. In particular, there is x with x g C2, x + <p„(w) g C2 so we must have m(w, x)(A") > 1/4; m(w, x + <¡>n(w))(B") > 1/4. Since n > 10, there are at least two distinct indices i, j < zzwith x + w,, x + wyg A". Let «j, zz2 with x + w, G A„, x + vi>-g /4n . We can assume zz1< zz2. There are also distinct indices k, I < zzwith * + WA + 4>„(W) ^ ß„,< * + W, + <|)„(w) G £ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use INVARIANT MEANS ON AN IDEAL 271 where n3 < nA. Assume, for example, nx < n3 (the case nx > n3 is similar). Then either i ^= k or i ^ I, say i i= k for definiteness. So we have x + w,■e An¡, x + wk + <i>„(w) G Bn>, so B,„^{Ani + wk-w, + 4,„(w))* 0. Note that nx > n. Since (p„(w) g <p'^(w) + V"' this contradicts the fact that Bll} n Dn = 0, and finishes the proof. 6. Proof of Theorem 5. This proof relies on Proposition 4. Let G be a compact, abelian, nondiscrete group. Then there exists a sequence (An) of sets of G with \An\ < 1/2 + 2'4 such that, for each r,,... ,f G G, there exists n with \C\i^ntj + An\ > 1/2. Proof. If a quotient of G has this property, so does G. The result is true for R/Z by taking An = \Jk k/n + [0,9/16«]. A similar idea works if G has cyclic quotients of arbitrarily larger order. If each element of G has a bounded order, it is enough to take for An the family of sets f>_1(^) where B c 77, H is a finite quotient of G, 4>is the quotient morphism and 1/2 < \B\ < 9/16. Proof of Theorem 5. The argument given at the beginning of the proof of Theorem 2 shows that one can suppose that, for some/ g Lx, Y = {p g Mq; p(f) = a}, where (25) a = sup{p(/),p G Mq). Proposition 2 still holds when 1/4 and 3/4 are replaced by 2~4 and 1 — 2~4. Let (k(n), /(«)) be an enumeration of N2. We can construct by induction a sequence (Bn) of disjoint sets of G, with \B„\ < 2""~5 such that for each sequence w = (wx,... ,wk(n)) of G, there exist two sequences u, v of G, there exists a measurable set F, with G\F g Q, and there exists r¡ > 0 such that the set {x g G; m(u, x)(xFf) > a - -q} can be covered by finitely many translates of the set {x g G; m(w ■v, x)(xFf) > a - 1/n; m(w ■v, x)(xBn) > 1 - 2"4}. For / g N, let C, = U{Bn; l(n) = I}. An argument used several times shows that there is p, g Mq, p¡(f) = a, Pi(C¡) > 1 — 2~4. From (25) we can assume that P/ g Eq. Let (An) be a sequence as in Proposition 4. Let A = U„ Bn n /4/(B). Since Q n ^4 = C, n /Í, we have |p/(/4) - P/(^4/)| < 2"4. Since p.,(A,) < |i4,| < 1/2 + 2"4 we get p¡(A) < 1/2 + 2'\ Now let tx,...,tq g G. There exists / with \À, n n^^r,-^/! > 1/2. In particular, for i ^ q we have |^°, n (i, + À,)\ > 1/2 so p(y4, n (i,. + >!/)) > 1/2. Since (>l n(z, + ^))A(^l/ n(/,.+ >!,)) c (cxC/Juí/.+ÍCXC,)), we get p,(A n(t, + A)) > p.,^, n(z, + ^,)) - 2"3 > 1/2 - 2"3. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use MICHEL TALAGRAND 272 By compacity, it follows that there is a cluster point p of the sequence (p,) with p g Y, p(A) < 1/2 + 2"3, and p(A n (t + /F)) > 1/2 - 2^3 for each t g G. Since 1/2 — 2"3 > (1/2 + 2"3)2, it then follows from [2] that p is not extremal. However, p g Ëq. Q.E.D. References 1. 2. 3. 4. E. M. Alfsen, Compact convex sets and boundary integrals. Springer-Verlag, Berlin, 1971. N. Bourbaki, Topologie générale. Chapitre 7, Hermann, Paris, 1947. G. Choquet, Lectures on analysis, Benjamin, New York, 1969. G. Converse, I. Namioka and R. R. Phelps, Extreme invariant positive operators. Trans. Amer. Math. Soc. 131 (1966), 376-385. 5. W. R. Emerson and F. P. Greenleaf, groups. Math. Z. 102 (1967), 370-384. 6. E. Granirer, No. 123 (1972). Covering properties and Folner conditions for locally compact Exposed points of convex sets and weak sequential convergence, Mem. Amer. Math. Soc. 7. F. P. Greenleaf, Invariant means on topological groups. Van Nostrand Nostrand, Princeton, N. J., 1969. Math. Stud., No. 16, Van 8. P. A. Meyer, Limites médiates, d'après Mokobodzki, Séminaire de Probabilités VII (Université de Strasbourg, 1973), Lecture Notes in Math., vol. 321, Springer-Verlag, Berlin and New York. 9. J. M. Rosenblatt, Invariant means and invariant ideals in LX(G) for a locally compact group G, J. Funct. Anal. 21 (1976), 31-51. 10. M. Talagrand, Géométrie des Simplexes de moyennes invariantes, J. Funct. Anal. 34 (1979), 304-337. 11. _, Moyennes invariantes s'annulant sur des idéaux, Compositio Math. 42 (1981), 213-216. Equipe d'Analyse - Tour 46, Université Paris VI, 75230 Paris Cedex 05, France Department of Mathematics, Ohio State University, Columbus, Ohio 43210 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use