Haar integral - Imported Knowlpedia Content

Definition

Let $G$ be a Hausdorff locally compact group, and let $C_c(G)$ denote the complex-valued continuous functions on $G$ with compact support. A left Haar integral is a nonzero positive linear functional

I:C_c(G)\longrightarrow\mathbb C

such that $I(L_gf)=I(f)$ for every $g\in G$ , where $L_gf(x)=f(g^{-1}x)$ . Haar’s theorem asserts that such a functional exists and is unique up to multiplication by a positive scalar. By measure representation, it has the form

I(f)=\int_G f\,d\mu

for a left Haar measure $\mu$ , so the functional and measure formulations are equivalent.

Positivity and normalization

Positivity means $I(f)\geq 0$ whenever $f$ is real-valued and nonnegative. It forces compatibility with monotone approximation and makes $I$ an actual integral, rather than merely an invariant linear functional. The uniqueness statement leaves one scale factor: choosing a normalization amounts to choosing that factor.

If $G$ is compact, one usually normalizes $I(1)=1$ . If $G$ is discrete, counting measure gives

I(f)=\sum_{x\in G}f(x)

for finitely supported $f$ . On $\mathbb R^n$ under addition, Lebesgue integration is a Haar integral.

Left and right invariance

Replacing left translations by $R_gf(x)=f(xg)$ defines a right Haar integral. Every locally compact group has both kinds. They coincide up to scale exactly in the unimodular case; in general, the modular function records how a left Haar integral changes under right translation.

The functional construction, representation by invariant measures, and uniqueness theorem are presented in Hewitt and Ross, “Invariant Functionals”.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. Routledge publisher record. Relevant: Chapter 2, “Haar Measure.”
Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis, Volume I, Springer, 1963. Springer DOI record. Relevant: “Invariant Functionals.”