---
tag: FCNQP0A60D
kind: concept
name: Quasi-invariant measure
msc: [28C10, 28D05, 22D40]
sources:
  - url: https://en.wikipedia.org/wiki/Quasi-invariant_measure
same_as:
  wikidata: Q7269454
  wikipedia: Quasi-invariant measure
---

## Summary

A [[measure|PR11BZQQPG]] is quasi-invariant under a [[group action|0WCY42EV9V]] if moving by any group element keeps the same [[null sets|VQQM4Y550T]].
The measure itself may change, but only by a density.
This weaker form of invariance is what one finds on homogeneous spaces without invariant measures and on infinite-dimensional spaces.

## Explanations

### L1 human

An invariant measure gives the same size to a set and to each of its moved copies.
A quasi-invariant measure only promises that a set of size zero stays of size zero when moved.
For example, the bell-curve measure on the real line is not invariant under shifts, since it puts more mass near $0$.
But a shifted set has zero bell-curve measure exactly when it has zero length, so the measure is quasi-invariant under shifts.