---
tag: 63TX60RTRQ
kind: concept
name: Modular function
msc: ["22D05", "28C10", "43A05"]
sources:
  - url: https://en.wikipedia.org/wiki/Haar_measure
---

## Summary

The modular function of a [[locally compact group|AHV05ZBCYW]] measures how far a left [[Haar measure|VY0HXD3GC4]] is from being right-invariant.
It is a continuous homomorphism $\Delta$ from the group to the multiplicative group of positive reals.
It appears in every formula that moves a Haar integral across a right translation or an inversion.
It is identically $1$ exactly for unimodular groups.

## Explanations

### L1 human

A left Haar measure does not change when a set is moved on the left, but it may grow or shrink when the set is moved on the right.
The growth factor depends only on the element used to move, and that factor is the modular function.
For the group of maps $x \mapsto ax + b$ with $a > 0$, moving a set on the right by such a map multiplies its left Haar measure by $1/a$.
For commutative groups left and right are the same, so the factor is always $1$.