Differentiation of an operator valued measure in a Banach rigging

Abstract

Remark 4. Let G be an infinite locally compact group, exPcG the subspace of all nonempty compact subsets of G. Define a mapping p :exp~O~-+~(O) taking each compact subset K to the subgroup topologically generated by K. Clearly, c(exp~G}~ ciexp G)~c~O) and p [exp~G)= (G). Therefore, if p is a continuous mapping, then c(g(G)) = c(G). It can be shown that p is continuous if and only if G is zero-dimensional and any noncompact topologically compactly generated subgroup of G is open. In particular, if G is zero-dimensional and inductively compact, then p is continuous. This yields a simpler proof of Theorem 2, but with the added assumption that G is zero-dimensional.