Haar measure

A Haar measure is a nonzero regular Borel measure on a locally compact group that is invariant under left or right translation. It is unique up to multiplication by a positive constant.

Locally compact group: A topological group in which every point has a compact neighborhood. Local compactness is the condition that guarantees existence of Haar measure.
Left translation: The map sending each group element x to gx for a fixed element g. Left Haar measure assigns the same measure to a set and each of its left translates.
Harmonic analysis on locally compact groups: The study of functions and representations on locally compact groups using invariant integration. Its integrals, norms, and Fourier transforms rely on Haar measure.
Left Haar measure: A nonzero regular Borel measure invariant under every left translation on a locally compact group. It is one convention covered by the existence and uniqueness theorem.
Borel measure: A measure defined on the Borel sets generated by a topological space’s open sets. Haar measure assigns sizes to Borel sets of the group.
Right translation: The map sending each group element x to xg for a fixed element g. Right Haar measure is invariant under these translates instead.
Fourier transform on locally compact abelian groups: A transform expressing functions on a locally compact abelian group through its characters. Haar integration defines the transform and its duality theory.
Right Haar measure: A nonzero regular Borel measure invariant under every right translation on a locally compact group. It is the alternative convention, related to left Haar measure by inversion.
Regular measure: A measure approximable from outside by open sets and from inside by compact sets. Regularity is part of the standard Haar measure theorem.
Integration on groups: Integration of functions over a topological group with respect to a measure on that group. Haar measure supplies the canonical translation-invariant integral.

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