Let (X , A) be a measurable space, and A0 a -field contained in A. Suppose that for any function T , the -field B

Let (X , A) be a measurable space, and A0 a σ-field contained in A.

Suppose that for any function T , the σ-field B is taken as the totality of sets B such that T −1(B) ∈ A. Then it is not necessarily true that there exists a function T such that T −1(B) ∈ A0. [An example is furnished by any A0 such that for all x the set consisting of the single point x is in A0.]