Haar Measure - Haar's Theorem

Haar's Theorem

There is, up to a positive multiplicative constant, a unique countably additive, nontrivial measure μ on the Borel subsets of G satisfying the following properties:

• The measure μ is left-translation-invariant: μ(gE) = μ(E) for every g in G and Borel set E.

• The measure μ is finite on every compact set: μ(K) < ∞ for all compact K

• The measure μ is outer regular on Borel sets E:

• The measure μ is inner regular on open Borel sets E:

Such a measure on G is called a left Haar measure. It can be shown as a consequence of the above properties that μ(U) > 0 for every non-empty open Borel subset U. In particular, if G is compact then μ(G) is finite and positive, so we can uniquely specify a left Haar measure on G by adding the normalization condition μ(G) = 1.

The left Haar measure satisfies the inner regularity condition for all σ-finite Borel sets, but may not be inner regular for all Borel sets. For example, the product of the unit circle (with its usual topology) and the real line with the discrete topology is a locally compact group with the product topology and Haar measure on this group is not inner regular for the closed subset {1} x . (Compact subsets of this vertical segment are finite sets and points have measure 0, so the measure of any compact subset of this vertical segment is 0. But, using outer regularity, one can show the segment has infinite measure.)

The existence and uniqueness (up to scaling) of a left Haar measure was first proven in full generality by André Weil. Weil's proof used the axiom of choice and Henri Cartan furnished a proof which avoided its use. Cartan's proof also proves the existence and the uniqueness simultaneously. A simplified and complete account of Cartan's argument was given by Alfsen in 1963. The special case of invariant measure for second countable locally compact groups had been shown by Haar in 1933.