Measure Theory pp 297-327 | Cite as

Haar Measure

  • Donald L. Cohn


A topological group is a set G that has the structure of a group (say with group operation (x,y) → xy) and of a topological space, and is such that the operations (x,y) → xy and xx −1 are continuous. Note that (x,y) → xy is a function from the product space G × G to G, and that we are requiring that it be continuous with respect to the product topology on G × G; thus xy must be “jointly continuous” in x and y, and not merely continuous in x with y held fixed and continuous in y with x held fixed (see Exercise 3). A locally compact topological group, or simply a locally compact group, is a topological group whose topology is locally compact and Hausdorff. A compact group is a topological group whose topology is compact and Hausdorff.


Open Subset Compact Subset Open Neighborhood Topological Group Compact Group 
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  1. The history of Haar measure is summarized in the notes at the ends of Sections 15 and 16 of Hewitt and Ross [41].Google Scholar
  2. The reader can find a more extensive introduction to topological groups in Pontryagin [69] or in Hewitt and Ross [41].Google Scholar
  3. The proof given here for the existence of Haar measure (which is a modification of Halmos’s modification of Weil’s [86] proof) depends on the axiom of choice. Proofs that do not depend on this axiom have been given by Cartan [16] and by Bredon [12], Cartan’s proof is given by Hewitt and Ross [41] and by Nachbin [65]. Hewitt and Ross and Nachbin also give calculations of Haar measure for a number of groups.Google Scholar

Copyright information

© Springer Science+Business Media New York 1980

Authors and Affiliations

  • Donald L. Cohn
    • 1
  1. 1.Department of MathematicsSuffolk UniversityBostonUSA

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