Published online by Cambridge University Press: 12 September 2009
Lebesgue proposed his view on integration in a short note (1901) and a famous book (1904). It was then a natural question to know whether Lebesgue's measure could be extended as a finitely additive measure defined on all subsets of Rn which is invariant under isometric transformations. Hausdorff answered negatively for n ≥ 3 in 1914 and Banach positively for n ≤ 2 in 1923. Von Neumann showed in 1929 that the deep reason for this difference lies in the group of isometries of Rn (viewed as a discrete group) which is amenable for n ≤ 2 and which is not so for n ≥ 3. In 1950, Dixmier extended the notion of amenability to topological groups [Dixmi–50].
In Section G.1, amenability is defined for a topological group in two equivalent ways, by the existence of an invariant mean on an appropriate space and as a fixed point property. Examples are given in Section G.2. For a locally compact group G, it is shown in Section G.3 that G is amenable if and only if 1G ≺ λG; a consequence is that amenability is inherited by closed subgroups of locally compact groups.
It is remarkable that amenability can be given a very large number of equivalent definitions, at least for locally compact groups. In Section G.4, we give Kesten's characterisation in terms of appropriate operators, which is crucial for the study of random walks on groups.