A left Haar measure on a locally compact group that is a Hausdorff space is a nonzero regular Borel measure, finite on compact sets, satisfying for every Borel set . Such a measure exists and is unique up to a positive scalar. Right Haar measure instead satisfies ; the two need not coincide. On a compact group, normalization gives total mass one and invariance on both sides. See Tornier, Haar Measures, Definition 2.1 and Theorem 2.2.
For a finite group, normalized Haar measure is uniform counting; for the circle group, it is normalized angle integration on the circle. These examples turn symmetry into a rule for averaging.
The Haar measure is an important concept in the area of harmonic analysis and abstract algebra, specifically in the context of topological groups. It is a way of defining a measure on a locally compact topological group that is left-invariant (or right-invariant), which means it remains unchanged (invariant) under the group's operations.
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