Haar measure
/ HAAR (Alfréd Haar) /
To average over a group — to symmetrize a function, project onto invariants, or define an inner product that no symmetry can disturb — you need a notion of 'volume' on the group that the group's own translations do not change. On the circle that is just arc length, unaffected by rotation. Haar measure is the existence-and-uniqueness theorem that such an invariant volume exists on essentially every group, making 'averaging over the group' a rigorous operation.
On a locally compact (Hausdorff) topological group G there exists a nonzero Radon measure mu that is left-invariant: mu(gE) = mu(E) for every g and every measurable set E (equivalently integral of f(g x) dmu(x) = integral of f(x) dmu(x)). This left Haar measure is unique up to a positive scalar. Symmetrically there is a right Haar measure. The two need not coincide; their discrepancy is recorded by the modular function Delta: G -> (0, infinity), a homomorphism with mu_right = Delta * mu_left. A group is called unimodular when Delta = 1, i.e. left and right Haar measures agree (giving a genuinely bi-invariant measure). All compact groups, all abelian groups, and all semisimple Lie groups are unimodular; the affine 'ax + b' group is the classic non-unimodular example. On a Lie group the Haar measure is concretely the volume form obtained by left-translating any fixed nonzero element of the top exterior power of g (the Maurer-Cartan frame), i.e. a left-invariant top-degree differential form.
Haar measure is the indispensable averaging tool: it is exactly what lets you average a representation to make it unitary (proving complete reducibility for compact groups), build invariant inner products, define characters and the Fourier/Plancherel theory, and integrate over G in the Peter-Weyl theorem. For a compact group one normalizes total mass to 1, turning averaging into a probabilistic 'expectation over the group.' Two honesty notes: existence is for LOCALLY COMPACT groups (the construction fails for infinite-dimensional groups, where typically no translation-invariant measure exists), and left and right invariance coincide only for unimodular groups — so 'the' Haar measure is bi-invariant only when the group is unimodular.
Averaging trick on a compact group G: given any representation on a vector space with some inner product <.,.>_0, define <u, v> = integral over G of <g.u, g.v>_0 dmu(g) using normalized Haar measure. Left-invariance of mu makes <.,.> G-invariant, so every finite-dimensional representation of a compact group is unitary — which forces complete reducibility (the orthogonal complement of an invariant subspace is invariant).
Averaging an inner product against Haar measure makes any compact-group representation unitary.
Left and right Haar measure agree only for unimodular groups (the modular function Delta = 1); the affine ax+b group is not unimodular. Existence requires local compactness — infinite-dimensional groups generally have no invariant measure.