An invariant of finitary codes with finite expected square root coding length

Let p and q be probability vectors with the same entropy h. Denote by B(p) the Bernoulli shift indexed by Z with marginal distribution p. Suppose that ' is a measure preserving homomorphism from B(p) to B(q). We prove that if the coding length of ' has a finite 1/2 moment, then � 2 = � 2 , where � 2 p = P i pi(−logpi − h)