Conditional mutual information by a same-marginal product law

Assume AC. Let X,Y,Z be random elements with standard-Borel value spaces and write $P_{ZXY}$ for the joint law of $(Z,X,Y)$ in that coordinate order. Let $K(z,\cdot )$ and $L(z,\cdot )$ be regular conditional laws of X and Y given Z=z. Let R be the probability law on the same product space whose rectangle values are $R(C\times A\times B)={\int }_{C}K(z,A)L(z,B)P_Z(dz)\mathrm{.}$ Its existence, uniqueness, and independence of the versions K,L are established in The same-marginal conditional-product law is well-defined. Define the conditional mutual information by $I(X;Y\mid Z):=D(P_{ZXY}\mathrm{\parallel }R)\in [0,+\mathrm{\infty }]\mathrm{.}$ The reference R has the same Z marginal as the joint law and makes X,Y independent conditionally on Z. The logarithm is natural, so finite values are measured in nats under Information units and logarithm-base convention. The definition is a divergence between full probability laws; it does not subtract two possibly infinite entropies.