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The number of extensions of an invariant mean

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The number of extensions of an invariant mean COMPOSITIO MATHEMATICA JOSEPHMAXROSENBLATT The number of extensions of an invariantmean Compositio Mathematica, tome 33, no 2 (1976), p. 147-159. <http://www.numdam.org/item?id=CM_1976__33_2_147_0> © Foundation Compositio Mathematica, 1976, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 147 THE NUMBER OF EXTENSIONS OF AN INVARIANT MEAN Joseph Max Rosenblatt COMPOSITIO MATHEMATICA, Vol. 33, Fasc. 2, 1976, pag. 147-159 Noordhoff International Publishing Printed in the Netherlands Abstract In a non-discrete 03C3-compact locally compact metric group G, a Baire category argument gives a continuum of measurable sets {Ay : y E FI independent on the open sets. If G is amenable as a discrete group and ~ : 0393 ~ {0, 1}, then any invariant mean 0 on CB(G) can be extended to an invariant mean 0 on L-(G) in such a way that for all y E F, 8(XA-y) = E(y). In a discrete group with card (G) &#x3E; No, if one is given an invariant subspace S of ém(G) with certain properties, then the axiom of choice gives a family {Ay : y e FI with card (F) = 2card(G) which is independent of S. If G is amenable as a discrete group and E : : 0393~ {0, 1}, then any invariant mean 0 on S can be extended to an invariant mean 0 on ém(G) in such a way that for all y E F, 8(XAy) = E(y). 0 Two different cases in which any invariant mean on an invariant subspace has many different extensions to an invariant mean on the whole space are considered here. In both cases, one constructs subs

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