# Haar measure

## Contents

## Definition

Let be a locally compact group.

### Left Haar measure

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

Condition name | Explanation |
---|---|

Borel subset | A subset in the -algebra generated by open subsets of |

Left-translation-invariant measure | If is measurable, and , then is measurable, and , where |

Countably additive (part of the usual definition of measure) | If are all measurable sets that are pairwise disjoint, and their union is , then is the sum of the values . |

Regular Borel measure | is finite for any compact subset . Also, for every Borel subset , and |

Nontrivial measure | for any nonempty open subset of |

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 .

### Right Haar measure

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

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 , define a correpsonding right Haar measure as:

where

This definition makes sense because is Borel if and only if is. The left Haar measure conditions on give rise to the right Haar measures on .

## 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 for any measurable subset . - 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 .

### 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.