= Compact group
{wiki}
= Compact topological group
{synonym}
A <compact group> is a <topological group> whose underlying <topological space> is a <compact Hausdorff space>. The <circle group> is a <compact group>: the <unit circle> is a closed bounded subset of the <complex plane>, so the <Heine-Borel theorem> gives <compactness>. The additive <group> $(\mathbb R,+)$ is a <locally compact group> and a <Hausdorff space>, but is not a <compact group>: the <open cover> $\{(-n,n):n\geq1\}$ has no finite subcover. A <finite group> with the <discrete topology> is another <compact group>; <compactness> does not require a smooth <manifold> structure.
The great advantage of a <compact group> is that one can average over every element without choosing a preferred origin. Precisely, every <locally compact group> that is a <Hausdorff space> admits a nonzero left <Haar measure> $\mu$, a regular <Borel measure> finite on <compact sets>, unique up to a positive scalar. Left invariance means $\mu(gA)=\mu(A)$ for every <Borel set> $A$ and every $g\in G$. There is also a right <Haar measure>; in general left and right <Haar measures> can differ. On a <compact group>, $0<\mu(G)<\infty$, and normalizing to $\mu(G)=1$ gives a unique regular <Borel probability measure> invariant under both left and right translation. See https://arxiv.org/html/2006.10956v1#S2[Tornier, Haar Measures, Theorem 2.2] for existence and uniqueness, and https://math.soimeme.org/~arunram/Teaching/1996padicGroupsSydney/TopologicalGroupsAndHaarMeasure.html[Ram's notes on left and right Haar measure] for the distinction.
For a <finite group> $G$, normalized <Haar measure> assigns each element mass $1/|G|$, so <integration> is simply $|G|^{-1}\sum_{g\in G}f(g)$. For the <circle group>, the corresponding <integral> is uniform angle averaging, computed below. On $(\mathbb R,+)$, <Lebesgue measure> is a <Haar measure>, but its total mass is infinite: no positive scalar multiple makes it a <probability measure>. Thus the existence of <Haar measure> does not make every <topological group> a <compact group>.
Historically, Alfréd Haar introduced the general invariant-measure construction in https://doi.org/10.2307/1968346[Der Massbegriff in der Theorie der kontinuierlichen Gruppen, Annals of Mathematics 34 (1933), pp. 147-169]. This bibliographic identification is also recorded in https://proceedings.centre-mersenne.org/articles/10.5802/mbk.21/[André Weil's archived seminar notes, Mesure de Haar, reference Haa33]. The conceptual step is to let <group> symmetry itself determine the weights used in <integration>, rather than relying on coordinates in an ambient <Euclidean space>.
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