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Do you regard finite groups as compact or discrete?

11 December, 2020

Every locally compact group admits a so-called Haar measure: this is a positive Radon measure on the Borel sigma-algebra of the group, which is invariant under left translations. These conditions determine the Haar measure uniquely up to a choice of positive scaling constant.

Examples of locally compact groups include compact groups and discrete groups.

  1. What’s the “natural” normalization of Haar measure for compact groups?
  2. What’s the “natural” normalization of Haar measure for discrete groups?

(This post brought to you after a panicked hour yesterday discovering compensating errors in a formula/proof, and a tedious couple of hours today spent rederiving the formulas for Fourier transform and Fourier inversion for G \times G when G is a finite group. It turns out that the identity 1^2=1 is rather dangerous…)

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