measurepreserving
1 Definition
Definition  Let $({X}_{1},{\U0001d505}_{1},{\mu}_{1})$ and $({X}_{2},{\U0001d505}_{2},{\mu}_{2})$ be measure spaces^{}, and $T:{X}_{1}\to {X}_{2}$ be a measurable transformation^{}. The transformation $T$ is said to be measurepreserving if for all $A\in {\U0001d505}_{2}$ we have that
$${\mu}_{1}({T}^{1}(A))={\mu}_{2}(A),$$ 
where ${T}^{1}(A)$ is, as usual, the set of points $x\in {X}_{1}$ such that $T(x)\in A$.
Additional Notation:

•
If $T$ is bijective^{}, measurepreserving, and its inverse^{} ${T}^{1}$ is also measurepreserving, then $T$ is said to be an measurepreserving transformation.

•
Measurepreserving transformations between the same measure space are sometimes called of the measure space.
Remarks:

•
The fact that a map $T:{X}_{1}\u27f6{X}_{2}$ is measurepreserving depends heavily on the sigmaalgebras ${\U0001d505}_{i}$ and measures ${\mu}_{i}$ involved. If other measures or sigmaalgebras are also in consideration, one should make clear to which measure space is $T:{X}_{1}\u27f6{X}_{2}$ measurepreserving.
 •
2 Properties

•
The composition of measurepreserving maps is again measurepreserving. Of course, we are supposing that the domains and codomains of the maps are such that the composition is possible.

•
Let $({X}_{1},{\U0001d505}_{1},{\mu}_{1})$ and $({X}_{2},{\U0001d505}_{2},{\mu}_{2})$ be measure spaces and $({X}_{1},\overline{{\U0001d505}_{1}},\overline{{\mu}_{1}})$ and $({X}_{2},\overline{{\U0001d505}_{2}},\overline{{\mu}_{2}})$ their completions. If $T:({X}_{1},{\U0001d505}_{1},{\mu}_{1})\u27f6({X}_{2},{\U0001d505}_{2},{\mu}_{2})$ is measurepreserving, then so is $T:({X}_{1},\overline{{\U0001d505}_{1}},\overline{{\mu}_{1}})\u27f6({X}_{2},\overline{{\U0001d505}_{2}},\overline{{\mu}_{2}})$.

•
Let $({X}_{1},{\U0001d505}_{1},{\mu}_{1})$ and $({X}_{2},{\U0001d505}_{2},{\mu}_{2})$ be measure spaces and ${T}_{1}:{X}_{1}\u27f6{X}_{1}$, ${T}_{2}:{X}_{2}\u27f6{X}_{2}$ be measurepreserving maps. Then, the product map ${T}_{1}\times {T}_{2}:{X}_{1}\times {X}_{2}\u27f6{X}_{1}\times {X}_{2}$, defined by
${T}_{1}\times {T}_{2}({x}_{1},{x}_{2}):=({T}_{1}({x}_{1}),{T}_{2}({x}_{2}))$ is a measurepreserving transformation of $({T}_{1}\times {T}_{2},{\U0001d505}_{1}\times {\U0001d505}_{1},{\mu}_{1}\times {\mu}_{2})$.
3 Examples

•
The identity map of a measure space $(X,\U0001d505,\mu )$ is always measurepreserving.

•
Let $G$ be a locally compact group (http://planetmath.org/TopologicalGroup). For each $a\in G$, the transformation $T(g):=ag$ is measurepreserving relatively to any left Haar measure. Similarly, any right translation^{} on $G$ any right Haar measure.

•
Every continuous surjective homomorphism^{} between compact Hausdorff is measurepreserving relatively to the normalized Haar measure (see this entry (http://planetmath.org/ContinuousEpimorphismOfCompactGroupsPreservesHaarMeasure)).
Title  measurepreserving 
Canonical name  Measurepreserving 
Date of creation  20130322 12:19:41 
Last modified on  20130322 12:19:41 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  17 
Author  asteroid (17536) 
Entry type  Definition 
Classification  msc 28D05 
Classification  msc 37A05 
Synonym  measure preserving 
Synonym  measurepreserving transformation 
Synonym  measurepreserving map 
Related topic  ErgodicTransformation 
Defines  invertible measurepreserving transformation 
Defines  endomorphism of a measure space 