G.1 Invariant means in .NET Produce barcode 3 of 9 in .NET G.1 Invariant means

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G.1 Invariant means using .net framework toget code 39 extended in web,windows application VS.NET both an amenable group and a group with Property (T) if and only if it is compact. G.1 Invariant means Let X be a set. A rin barcode 39 for .NET g R of subsets of X is a non-empty class of subsets of X which is closed under the formation of union and differences of sets: if A, B R, then A B R and A \ B R.

De nition G.1.1 A mean m on a ring R of subsets of X containing X is a nitely additive probability measure on R, that is, m is a function from R to R with the following properties: (i) m(A) 0 for all A R; (ii) m(X ) = 1; (iii) m(A1 An ) = m(A1 ) + + m(An ) if A1 , .

. . , An R are pairwise disjoint.

If a group G acts on X leaving R invariant, then m is said to be a G-invariant mean if (iv) m(gA) = m(A) for all g G and A R. For a ring R of subsets of X , denote by E the vector space of complex-valued functions on X generated by the characteristic functions A of subsets A in R. There is a natural bijective correspondence between means m on R and linear functionals M on E such that m(A) = M ( A ) for all A R.

. In case R is G-invari ant for some group G acting on X , the mean m is G invariant if and only if M (g ) = M ( ) for all g G and E, where g is the function on X de ned by g (x) = (gx) for x X . Assume now that there is given a -algebra B of subsets of X and a measure on (X , B). There is again a bijective correspondence between appropriate linear functionals on L (X , ) and appropriate means on B, as we now explain.

De nition G.1.2 Let (X , B, ) be a measure space and let E be a closed subspace of L (X , B, ) which contains the constant functions and is closed under complex conjugation.

A mean on E is a linear functional M : E C with the following properties: (i) M (1X ) = 1; (ii) M ( ) 0 for all E with 0.. Amenability Let G be a group acti ng on E. We say that M is G-invariant if, moreover, (iii) M (g ) = M ( ) for all g G and E. Remark G.

1.3 (i) A mean M on E is automatically continuous. Indeed, .

1X. 1X .. Hence, . M ( ). by (i) and (ii Code 39 Full ASCII for .NET ). (ii) A mean M on L (X , B, ) de nes a mean m on the -algebra B by m(A) = M ( A ) for all A B.

Observe that m is absolutely continuous with respect to , in the sense that m(A) = 0 for all A B with (A) = 0. Conversely, if m is a mean on B which is absolutely continuous with respect to , then there exists a unique mean M on L (X , B, ) such that m(A) = M ( A ) for all A B. Indeed, de ne.

M ( ) =. i m(Ai ). if = m i Ai is a measurable simple function on X . By nite additivity i=1 of m, this de nition does not depend on the given representation of as linear combination of characteristic functions of measurable subsets. Let be a measurable bounded function on X .

There exists a sequence ( n )n of measurable simple functions on X converging uniformly on X to . It is easily veri ed that (M ( n ))n is a Cauchy sequence in C and that its limit does not depend on the particular choice of ( n )n . De ne then M ( ) = lim M ( n ).

. Since m is absolutely USS Code 39 for .NET continuous with respect to , the number M ( ) depends only on the equivalence class [ ] of in L (X , ), and we can de ne M ([ ]) = M ( ). One checks that M is a mean on L (X , ).

For more details, see [HewSt 69, (20.35) Theorem]. From now on, we will identify a mean M on a space E in the sense of De nition G.

1.2 and the corresponding mean m on B (or on R), and we will use the same notation m in both cases. Let G be a topological group.

Let (G) be the Banach space of bounded functions on G. The group G acts on (G) by left translations: g 1 , .
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