Invariant means on locally compact groups and transformation groups

This thesis deals with two separate questions in the area of invariant means on locally compact groups. Granirer has shown that, for certain discrete semigroups S, the range of a left invariant mean on the algebra m(S) is the entire [0,1] interval and further, that this range can be obtained on a nested family of left almost convergent subsets of S. We generalize the first part of his result to show that the range of every left invariant mean on L[sup ∞] (G) for a locally compact group G is [0,1]. If G is abelian we also show that this range is attained on a nested family of left almost convergent Borel subsets of G. In the last chapter we deal with the problem of extending the concept of amenability for a locally compact group G to the situation where G acts on the space G/H of left cosets of G with respect to a closed subgroup H ( a group acting in this way is called a transformation group). We introduce a definition of the amenable action of G on various closed subspaces of L[sup ∞] (G/H,ν) (ν a quasi-invariant measure on G/H) which is equivalent to the one given by Greenleaf but is obtained by different methods. We also prove analogues of several well-known theorems concerning the amenability of locally compact groups.Science, Faculty ofMathematics, Department ofGraduat