##  [Measure-Preserving Transformation](/measure-preserving-transformation) 

  ##  [Measure-Preserving Transformation](https://natural.quantumdictionary.io/measure-preserving-transformation-0) 

  

 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)



**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A measurable map T: X → X on a measure space (X, Σ, μ) that satisfies μ(T^{-1}(A)) = μ(A) for every measurable set A; equivalently, the pushforward measure T_*μ equals μ.