Measure Theory pp 250-265 | Cite as

Haar Measure

  • Paul R. Halmos
Part of the Graduate Texts in Mathematics book series (GTM, volume 18)


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.


Invariant Measure Compact Group Positive Measure Haar Measure Compact Hausdorff Space 
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© Springer Science+Business Media New York 1950

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  • Paul R. Halmos

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