Before beginning our investigation of measure theory in topological groups, we shall devote this brief section to the proof of two topological results which have important measure theoretic applications. The results concern full subgroups; a subgroup Z of a topological group X is full if it has a non empty interior. We shall show that a full subgroup Z of a topological group X embraces the entire topological character of X—everything in X that goes beyond Z is described by the left coset structure of Z which is topologically discrete. We shall show also that a locally compact topological group always has sufficiently small full subgroups—i.e. full subgroups in which none of the measure theoretic pathology of the infinite can occur.
KeywordsInvariant Measure Compact Group Positive Measure Haar Measure Compact Hausdorff Space
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