Definition:Factor of Measure-Preserving Dynamical System

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 $\struct {X, \BB, \mu, T}$ :
 * 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}$