Given a measure-preserving transformation T acting on a σ \sigma -finite measure space ( X , A , m ) (X,\mathcal {A},m) and a σ \sigma -finite sigma algebra B ⊂ A \mathcal {B} \subset \mathcal {A} , the conditional expectations E ( ⋅ | B ) E( \cdot |\mathcal {B}) acting on L ∞ ( A ) {L^\infty }(\mathcal {A}) and E ( ⋅ | T − 1 B ) E( \cdot |{T^{ - 1}}\mathcal {B}) acting on L ∞ ( T − 1 A ) {L^\infty }({T^{ - 1}}\mathcal {A}) are known to be related by the formula [ E ( f | B ) ] ∘ T = E ( f ∘ T | T − 1 B ) [E(f|\mathcal {B})] \circ T = E(f \circ T|{T^{ - 1}}\mathcal {B}) . In this note the conditional expectation E ( ⋅ | T − 1 B ) 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 ∘ T − 1 ) / d m d(m \circ {T^{ - 1}})/dm . It is precisely in the non-measure-preserving case that the measurability of d ( m ∘ T − 1 ) / d m d(m \circ {T^{ - 1}})/dm plays an important role. Relatedly, it is shown that if composition by T intertwines E ( ⋅ | B ) E( \cdot |\mathcal {B}) and any mapping Λ \Lambda , then Λ \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 ∗ {C^ \ast } -algebras, and the paper concludes with a brief description of this C ∗ {C^\ast } -algebra setting.