Haar Measure - Uses

Uses

Historically, the first use of the Haar theorem was the solution, by von Neumann, of Hilbert's fifth problem in the case of compact groups. In fact, von Neumann's article was published in the same issue of Annals of Mathematics as Haar's article and immediately after it.

The Haar measures are used in harmonic analysis on arbitrary locally compact groups; see Pontryagin duality. A frequently used technique for proving the existence of a Haar measure on a locally compact group G is showing the existence of a left invariant Radon measure on G.

In estimation theory, Haar measures can be used as non-informative priors, being Jeffreys priors for various questions. For instance, translation invariance of the (improper) uniform distribution on the real numbers (the Haar measure with respect to addition) corresponds to no information about location, and thus it is the Jeffreys prior for the unknown mean of a Gaussian distribution, the mean being a measure of location.

Unless G is a discrete group, it is impossible to define a countably-additive left invariant measure on all subsets of G, assuming the axiom of choice. See non-measurable sets.