Iterated conditional expectations

Consider the probability space ( [ 0 , 1 ) , B , λ ) , where B is the Borel σ-algebra on [ 0 , 1 ) and λ the Lebesgue measure. Let f = 1 [ 0 , 1 / 2 ) and g = 1 [ 1 / 2 , 1 ) . Then for any ε > 0 there exists a finite sequence of sub-σ-algebras G j ⊂ B ( j = 1 , … , N ) , such that putting f 0 = f and f j = E ( f j − 1 | G j ) , j = 1 , … , N , we have ‖ f N − g ‖ ∞ < ε ; here E ( ⋅ | G j ) denotes the operator of conditional expectation given σ-algebra G j . This is a particular case of a surprising result by Cherny and Grigoriev (2007) [1] in which f and g are arbitrary equidistributed bounded random variables on a nonatomic probability space. The proof given in Cherny and Grigoriev (2007) [1] is very complicated. The purpose of this note is to give a straightforward analytic proof of the above mentioned result, motivated by a simple geometric idea, and then show that the general result is implied by its special case.