By N. Piskunov
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Extra info for Calculo Diferencial e Integral – Tomo I
Example text
Given a finite part B of G, the intersection invariant with A. B(A) the inequality mD( ~ b-iA) beB which the proof of the first result: -I(A) mB'D'bo < = is true for any given element b 0 of B, leads quickly to the proof of the second result. 13. Example. e. in G that are strictly The greatest lower bound of all increments iA(T) a = strongly Denote to G\{a}, the set of - m(a~A) of G that do not contain also the limit of these a is, when A increments because m is subadditive. by ia(T) functions this limit; on T (depending as a greatest Because of the property bounded below.
2. Proposition. A subgroup Proof. We use here the ameaning of an amenable group is amenable. filter. For every finite part D of H and every positive real number exists a finite part A of G such that roD(A) =< ~. x i = HxinA. On the other hand, the set {xeA,~deD,dx~A} is the disjoint union of the sets {xeAi,~deD,dx~A i} because D is a subset of H. The two equalities IAI therefore simultaneously = ~ IAil and At least one of the Aix i belongs hold mD(A) to M(D,~) = mD(Aixi ) as well as its right trans- late Ai, which is a part of H.
Define an action of G on this convex set by setting Tg(x) = Ts(g)(x) Because G has the fixed-point iant point under property, there exists at least one invar- this action of G, which is then an invariant point under the given action of H. 2. Proposition. A subgroup Proof. We use here the ameaning of an amenable group is amenable. filter. For every finite part D of H and every positive real number exists a finite part A of G such that roD(A) =< ~. x i = HxinA. On the other hand, the set {xeA,~deD,dx~A} is the disjoint union of the sets {xeAi,~deD,dx~A i} because D is a subset of H.