= Haar measure
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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 $\mu(gA)=\mu(A)$ for every <Borel set> $A$. Such a <measure> exists and is unique up to a positive scalar. Right <Haar measure> instead satisfies $\mu(Ag)=\mu(A)$; the two need not coincide. On a <compact group>, normalization gives total mass one and invariance on both sides. See https://arxiv.org/html/2006.10956v1#S2[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.
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