Let $\widetilde{\Omega}=(\Omega, P, \mathcal A)$ and $\widetilde{\Sigma}=(\Sigma, Q, \mathcal B)$ be probability measure spaces, and let $\phi:\Omega\to\Sigma$ and $\psi:\Sigma\to\Omega$ be measurability preserving maps. The maps $\phi$ and $\psi$ induce $\phi^{-1}:\mathcal B\to\mathcal A$ and $\psi^{-1}:\mathcal A\to\mathcal B$. By $(\mathcal A,\mathcal A,f)$ we denote the Chu space associated with the probability measure space $(\Omega,P,\mathcal A)$. Our main results are:

Theorem 1. Let $P(\widetilde{\Omega})=(\mathcal A, \mathcal A, f)$ and $P(\widetilde{\Sigma})=(\mathcal B,\mathcal B, g)$ be Chu spaces associated with $\widetilde{\Omega}$ and $\widetilde{\Sigma}$, respectively. If $\Phi=(\psi^{-1}, \phi^{-1}):P(\widetilde{\Omega})\to P(\widetilde{\Sigma})$ is a Chu morphism, then both $\phi$ and $\psi$ are measure preserving.

Theorem 2. The pair $(\phi, \psi)$ is mutually measure preserving if and only if $\Phi=(\psi^{-1}, \phi^{-1}):(\mathcal A,\mathcal A, f)\to (\mathcal B, \mathcal B, g)$ is a Chu morphism.