Relative Entropy of the Image under a Bimeasurable Bijection that Moves the Reference Measure by a Positive Density

In the settings of The Real Numbers: Standing Notation and Background and Measure Spaces and the Lebesgue Integral: Standing Notation, the measure space of the latter being instantiated at each use by the measure space named, let $(S,\mathcal{S})$ be a measurable space and let $\gamma$ and $\nu$ be probability measures on it. Densities with respect to $\gamma$ are those of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities; finite relative entropy with respect to $\gamma$ and $H(\cdot\,|\,\gamma)$ are those of that definition; $\log$ is the natural logarithm. Let $T:S\to S$ be a bijection such that $T$ and its inverse map $T^{-1}$ are measurable with respect to $\mathcal{S}$; for $B\subseteq S$ the preimage of $B$ under $T$ is then the image of $B$ under $T^{-1}$, and both are written $T^{-1}(B)$. For a probability measure $\lambda$ on $(S,\mathcal{S})$ let $T_{\#}\lambda$ be the image measure $B\mapsto\lambda(T^{-1}(B))$ of claim 1 of that lemma. Let $g:S\to\mathbb{R}$ be measurable with $0<g(s)$ for every $s\in S$, and suppose that $g$ is a density of $T_{\#}\gamma$ with respect to $\gamma$.

1. (Density of the image)¶ If $f$ is a density of $\nu$ with respect to $\gamma$, then $(f\circ T^{-1})\,g$ is a density of $T_{\#}\nu$ with respect to $\gamma$.

2. (Relative entropy of the image)¶ Suppose that $\nu$ has finite relative entropy with respect to $\gamma$ and that the function $s\mapsto\log g(T(s))$ is integrable with respect to $\nu$. Then $T_{\#}\nu$ has finite relative entropy with respect to $\gamma$, and

$$H(T_{\#}\nu\,|\,\gamma)=H(\nu\,|\,\gamma)+\int_{S}\log g(T(s))\,\nu(ds).$$