Let and be probability measure spaces, and let and be measurability preserving maps. The maps and induce and . By we denote the Chu space associated with the probability measure space . Our main results are:
Theorem 1. Let and be Chu spaces associated with and , respectively. If is a Chu morphism, then both and are measure preserving.
Theorem 2. The pair is mutually measure preserving if and only if is a Chu morphism.