Modular function of a locally compact group

Let $G$ be a locally compact group and let $\mu$ be a left Haar measure. The modular function is the unique map

\Delta_G:G\longrightarrow \mathbb R_{>0}

satisfying

\int_G f(xg)\,d\mu(x) = \Delta_G(g)^{-1}\int_G f(x)\,d\mu(x)

for every $g\in G$ and every compactly supported continuous $f$ . It is a continuous group homomorphism and does not depend on rescaling $\mu$ . Thus $\Delta_G$ measures precisely the failure of a left Haar measure to be invariant under right translations. The inverse in the displayed convention is important; some references define the reciprocal modular function.

Measure-theoretic meaning

For fixed $g$ , right translation sends $\mu$ to another left Haar measure, so uniqueness of Haar measure produces a positive scale factor. Those factors multiply under composition of right translations, which gives the homomorphism law

\Delta_G(gh)=\Delta_G(g)\Delta_G(h).

Inversion transports a left Haar measure to a right Haar measure. The modular function compares these two invariant-measure conventions.

Unimodularity

The group $G$ is unimodular exactly when $\Delta_G\equiv 1$ , equivalently when a left Haar measure is also right invariant. Abelian, compact, and discrete groups are unimodular. Nontrivial semidirect products supply many non-unimodular examples.

Role in harmonic analysis

The modular factor is required in the involution on a group convolution algebra and in unitary formulas involving right translation. It also enters induced representations and normalized parabolic induction. Ignoring it silently imports the unimodular case into formulas that are otherwise false.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. Routledge publisher record. Relevant: §2.4, “The Modular Function.”
Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis, Volume I, Springer, 1963. Springer DOI record. Relevant: invariant functionals, translations, and convolution.