Existence and uniqueness of Haar measure

Remarks

Such a measure is a left Haar measure of $G$. One proof compares the size of a compact set $K$ with the number of translates of a small open set needed to cover it. These ratios have a limit point, by a compactness argument, which gives an invariant content and then a Radon measure. Another proof builds an invariant positive linear functional on compactly supported continuous functions and applies the Riesz–Markov–Kakutani representation theorem. Uniqueness follows from a Fubini argument applied to $\mu \otimes \nu$ and the map $(x, y) \mapsto (x, xy)$.

Example: on $G = (\R^n, +)$, Lebesgue measure is a Haar measure, so every translation-invariant Radon measure on $\R^n$ is a multiple of Lebesgue measure. On the circle group, the Haar measures are the multiples of arc length. On a discrete group, the Haar measures are the multiples of counting measure.

Local compactness enters the proof through compact sets with nonempty interior, which give the comparison of sizes.

Going deeper

For a student

Haar measure lets one integrate over a group, which is the base of harmonic analysis on groups. It gives the group algebra $L^1(G)$, the Peter–Weyl theorem for compact groups and Pontryagin duality for abelian groups. For a compact group the measure is finite and is usually normalized to total mass $1$. Left and right Haar measures may differ, and their ratio is the modular function of $G$.