---
tag: F608XX51KA
kind: concept
name: Quasi-measure-preserving map
msc: [28D05, 37A40]
sources:
  - url: https://en.wikipedia.org/wiki/Quasi-invariant_measure
---

## Summary

A quasi-measure-preserving map is a measurable map between measure spaces under which the preimage of every null set is a null set.
Equivalently, the pushforward of the first measure is [[absolutely continuous|P0P21KDV6E]] with respect to the second one.
Such maps respect "almost everywhere" statements, so they act on equivalence classes of measurable functions.
Every [[measure-preserving transformation|556WBDAHNP]] is quasi-measure-preserving.

## Explanations

### L1 human

A measure-preserving map keeps the size of every set.
A quasi-measure-preserving map is allowed to change sizes, but it may not turn something of size zero into something of positive size when one pulls back.
For example, the map $x \mapsto 2x$ on the real line doubles lengths, so it does not preserve length.
Still, the preimage of a set of length zero has length zero, so the map is quasi-measure-preserving.