---
tag: CE0V9686M8
kind: definition
name: Haar measure
msc: ["28C10", "22D05", "43A05"]
defines: VY0HXD3GC4
uses: [AHV05ZBCYW, HZNEQQBCW4, 0WQXY32Q8D]
sources:
  - url: https://en.wikipedia.org/wiki/Haar_measure
---

## Definition

Let $G$ be a [[locally compact group|AHV05ZBCYW]] and let $\mathcal{B}$ be the $\sigma$-algebra of [[Borel sets|0WQXY32Q8D]] of $G$.
A *left Haar measure* on $G$ is a measure $\mu$ on $\mathcal{B}$ such that:

1. $\mu$ is a [[Radon measure|HZNEQQBCW4]]: it is finite on compact sets, outer regular on Borel sets and inner regular on open sets;
2. $\mu$ is not the zero measure;
3. $\mu$ is left-invariant: $\mu(gA) = \mu(A)$ for every $g \in G$ and every $A \in \mathcal{B}$, where $gA = \set{ga : a \in A}$.

A *right Haar measure* is defined in the same way with $\mu(Ag) = \mu(A)$ in clause 3.

## Remarks

Example: Lebesgue measure $\lambda$ on $(\R^n, +)$.
It is finite on compact sets, regular and nonzero, and $\lambda(x + A) = \lambda(A)$ for every $x \in \R^n$ and Borel set $A$.

Non-example: the Dirac measure $\delta_0$ on $(\R, +)$ is a nonzero Radon measure but fails clause 3, since $\delta_0(\set{0}) = 1$ and $\delta_0(1 + \set{0}) = \delta_0(\set{1}) = 0$.

In integral form, left invariance says $\int_G f(gx) \, d\mu(x) = \int_G f(x) \, d\mu(x)$ for every $g \in G$ and every compactly supported continuous $f$.
If $\mu$ is a left Haar measure, then $A \mapsto \mu(A^{-1})$ is a right Haar measure.
A left Haar measure gives positive measure to every nonempty open set.