On fixed points for semi-groups of linear operators

That such theorems, when applied to the space X of bounded functions on a set S, have measure-theoretic significance was noted in [3]. Let H be a semi-group of 1-1 transformations of S into S. To each aCH we correspond the transformation T, of X into itself where Ta(f) is the function f( (s)), s C S, f CX. These transformations form a semi-group G each element of which has norm one. Furthermore G has the unit function as a nonzero fixed point. If G* also possesses a nonzero fixed point x4, then there is a bounded additive set function A defined for all subsets of S such that