Mutual information for standard-Borel random elements

Assume AC. Let $X,Y$ be random elements with standard-Borel value spaces $(\mathsf{X},\mathcal{X})$ and $(\mathsf{Y},\mathcal{Y})$, and let $P_{XY}$, $P_X$, and $P_Y$ be their joint and marginal probability laws. The product law $P_X\otimes P_Y$ exists by For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique. Define the mutual information in nats by

$I_{\mathrm{n}\mathrm{a}\mathrm{t}}(X;Y):=D(P_{XY}\mathrm{\parallel }{P}_X\otimes P_Y)\in [0,+\mathrm{\infty }],$

where $D$ is the directional Kullback–Leibler divergence of Kullback–Leibler divergence. This definition remains meaningful when one or both laws have no Lebesgue density and does not subtract differential entropies that may be infinite.

The underlying spaces are standard Borel, as defined in Standard Borel spaces. AC is part of the definition's dependency contract through the Radon–Nikodym construction in the KL supplier.