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1 for invariant means on 03(0) in the lit erature . 1). For (=>) consider the set 2 of means on X = UCBTW), a weak * compact convex set in X*. Define Tw(y,, so < Tmm,f> a: for a: 6 0, m 6 X*, f6 X. Then this defines an affine group action of G on 2 and the map 0 x E a 2 is continuous since :13]. ,<, + 0 (this is not true in larger spaces of functions such as X x 08(0)). Hence there is a fixed point m e E and m is the desired LIM: m(zf) = < Tw 1(m),f > -= m(f) for all a: e G. Conversely, let m be any mean on X and let G act affinely, continuously on compact convex set S in E.

Urup and nrwups uyn - Hfllp- When ¢ 6 L1(G) is identified with a measure as above, this gives the usual formula ¢>*f(8) = y¢ * s) = /f(t"1s)¢(t)dt f*¢(s) f*u¢(8) /f(t)¢(t-1s)dt . For m c 0 write 806 for the point mass at m; then if f e LNG): 24 INVARIANT MEANS ON TOPOLOGICAL GROUPS 5m *f(s) f(w"1s) = wf(s) f*8m(s) = f(sw 1)A(w 1). If f c L°°(G ) there are special difficulties when G is not unimodular (923 e L1 does not => ¢' 5 L1); the convolutions which make sense are: ¢,*f(s) = /f(t-1s)¢(t)dt f*¢'(8) =/f(t)¢'(t ls)dt = /f(t)¢>(s"1t)dt where ¢ 5 L1(G).

1). } weak* convergent to our mean m then s]. ) 6 S for each index. 7 functions on S, so A(S) D {f*|S: f* e E*}; let rA be the weakest topology on S making these functions continuous. 7 Tllj s) (TA) >so for some so 6 S. Of course we still have u]. > m weak* in 2 C X*, so < f*, so > < f*, T#j(s) > / dpj(g) » /< f*, Tg(s) > dm(g) = < f*, Tm(s) > for all affine continuous functions f* e A(S), and in particular for all f* e E*. Thus Tm(s) = so 6 S. 52 INVARIANT MEANS ON TOPOLOGICAL GROUPS Now if f* e E* and if we write f*oTw(s) = < f*, Tw(s) >, then f*°Tm is obviously in A(S); thus Tm(s) is a fixed point if m is a LIM on X since: < f*, Tw(Tm(s)) > = < f*on, Tm(s) > = /< ora, Tg(s) > dm(g) = /< ft ng(s)> dm(g) = / < f*, rye) > dm(g) = < f*, rms) > all f* 6 E*.