Haar measure

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Definition

Let G be a locally compact group.

Left Haar measure

A left Haar measure is a left-translation-invariant countably additive regular nontrivial measure \mu on the Borel subsets of G. The conditions are explained below:

Condition name Explanation
Borel subset A subset in the \sigma-algebra generated by open subsets of G
Left-translation-invariant measure If S \subseteq G is measurable, and g \in G, then gS is measurable, and \mu(gS) = \mu(S), where gS = \{ gs \mid g \in G \}
Countably additive (part of the usual definition of measure) If S_1, S_2, \dots, S_n, \dots are all measurable sets that are pairwise disjoint, and their union is S, then \mu(S) is the sum of the values \mu(S_i).
Regular Borel measure \mu(K) is finite for any compact subset K. Also, for every Borel subset S, \mu(S) = \inf \{\mu(U), S \subseteq U, U \operatorname{open} \} and \mu(S) = \sup \{ \mu(UK), K \subseteq S, K \operatorname{compact} \}
Nontrivial measure \mu(U) > 0 for any nonempty open subset U of G

The left Haar measure for a locally compact group is unique up to scalar multiples, i.e., the quotient of any two left Haar measures is a scalar.

For a compact group, there is a unique choice of normalized Haar measure, i.e., a unique left Haar measure where the total measure of the group is 1.

Right Haar measure

A right Haar measure is a left-translation-invariant countably additive regular measure \mu on the Borel subsets of G.

The right Haar measure for a locally compact group is unique up to scalar multiples, i.e., the quotient of any two left Haar measures is a scalar.

Relationship between left and right Haar measure

Any left Haar measure on a group can be used to canonically define a right Haar measure: for a left Haar measure \mu_l, define a correpsonding right Haar measure as:

\mu_r(S) := \mu_l(S^{-1})

where S^{-1} = \{ s^{-1} \mid s \in S \}

This definition makes sense because S is Borel if and only if S^{-1} is. The left Haar measure conditions on \mu_l give rise to the right Haar measures on \mu_r.

Particular cases

Compact groups

For a compact group, the following additional features are true:

  • The left Haar measures coincide with the right Haar measures. In particular, we just talk of a Haar measure without talking of left or right. This Haar measure thus satisfies the additional condition that \mu(S) = \mu(S^{-1}) for any measurable subset S.
  • Further, since the total measure of the group is finite, there is a natural choice of normalized Haar measure where the total measure of the group is 1.

Bi-invariant Haar measures?

A bi-invariant Haar measure is a measure that is both a left and a right Haar measure. Not all locally compact groups have bi-invariant Haar measures, but compact groups, locally compact abelian groups, and some other examples of locally compact groups have bi-invariant Haar measures. If a group has a bi-invariant Haar measure, then every left Haar measure is a right Haar measure, and hence a bi-invariant Haar measure.