# Definition:Factor of Measure-Preserving Dynamical System

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## Definition

Let $\struct {X, \BB, \mu, T}$ and $\struct {Y, \CC, \nu, S}$ be measure-preserving dynamical systems.

Then $\struct {Y, \CC, \nu, S}$ is called a factor of $\struct {X, \BB, \mu, T}$ if and only if:

there exist $X' \in \BB$, $Y' \in \CC$ and a mapping $\phi : X' \to Y'$ such that:
$\map \mu {X'} =1$
$\map \mu {Y'} = 1$
$T \sqbrk {X'} \subseteq X'$
$T \sqbrk {Y'} \subseteq Y'$
$\phi$ is measure-preserving with respect to $\struct {X', \BB_{X'}, \mu \restriction {\BB_{X'} } }$ and $\struct {Y', \CC_{Y'}, \nu \restriction {\CC_{Y'} } }$
$\forall x \in X' : \map {\phi \circ T} x = \map {S \circ \phi} x$
where
$T \sqbrk {X'}$ denotes the image of $X'$ under $T$
$S \sqbrk {Y'}$ denotes the image of $Y'$ under $S$
$\BB_{X'}$ denotes the trace $\sigma$-algebra of $X'$ in $\BB$
$\CC_{Y'}$ denotes the trace $\sigma$-algebra of $Y'$ in $\CC$
$\mu \restriction {\BB_{X'} }$ denotes the restriction of $\mu$ to $\BB_{X'}$
$\nu \restriction {\CC_{Y'} }$ denotes the restriction of $\nu$ to $\BB_{Y'}$

## Also defined as

Ignoring null sets, we simply say $\phi : X \to Y$ is a measure-preserving mapping such that the diagram commutes:

$\begin{xy}\xymatrix@+1em@L+3px{ X \ar[r]^T \ar[d]_\phi & X \ar[d]^\phi \\ Y \ar[r]_S & Y }\end{xy}$