Definition:Left Haar Measure

Definition

Let $\struct {G, \odot, \tau}$ be a locally compact topological group.

Let $\BB$ be the Borel $\sigma$-algebra.

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A measure $m$ on $\BB$ is called a left Haar measure if and only if:

$(1): \quad \forall U \in \tau : U \ne \O \implies \map m U > 0$

$(2): \quad$ $m$ is left translation invariant, that is:

$\forall g \in G, \forall E \in \BB : \map m {g \odot E} = \map m E$

where:

$ g \odot E := \set {g \odot h : h \in E}$

Source of Name

This entry was named for Alfréd Haar.

Sources

• 1974: Paul R. Halmos: Measure Theory: Chapter XI: Haar Measure