Descriptions of conditional expectations induced by non-measure preserving transformations

Abstract:

Given a measure-preserving transformation T acting on a $\sigma$-finite measure space $(X,\mathcal {A},m)$ and a $\sigma$-finite sigma algebra $\mathcal {B} \subset \mathcal {A}$, the conditional expectations $E( \cdot |\mathcal {B})$ acting on ${L^\infty }(\mathcal {A})$ and $E( \cdot |{T^{ - 1}}\mathcal {B})$ acting on ${L^\infty }({T^{ - 1}}\mathcal {A})$ are known to be related by the formula $[E(f|\mathcal {B})] \circ T = E(f \circ T|{T^{ - 1}}\mathcal {B})$ . In this note the conditional expectation $E( \cdot |{T^{ - 1}}\mathcal {B})$ is investigated in the non-measure-preserving case, and those transformations for which the above equation holds are characterized in terms of measurability conditions for $d(m \circ {T^{ - 1}})/dm$. It is precisely in the non-measure-preserving case that the measurability of $d(m \circ {T^{ - 1}})/dm$ plays an important role. Relatedly, it is shown that if composition by T intertwines $E( \cdot |\mathcal {B})$ and any mapping $\Lambda$, then $\Lambda$ is a conditional expectation induced by a measure equivalent to m. These results were motivated by a result concerning induced conditional expectation operators on ${C^ \ast }$-algebras, and the paper concludes with a brief description of this ${C^\ast }$-algebra setting.