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Let $X$ be a measurable space, and let $T$ be a measurable transformation $T:X \to X$. Let $\mathcal{P}(X)$ be the space of probability measures on $X$, equipped with the weak* topology. Define the $T$-relative entropy by $h_T :\mathcal{P}(X) \to \mathbb{R}$ by $$h_T(\nu) = -\int_X\log \frac{dT^{-1}_*\nu}{d\nu}(x)d\nu(x).$$

Is $h_T$ continuous on the set in which it is finite? Is it upper semi-continuous?

Thanks!

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# Continuity of relative entropy with respect to the weak* topology

Let $X$ be a measurable space, and let $T$ be a measurable transformation $T:X \to X$. Let $\mathcal{P}(X)$ be the space of probability measures on $X$, equipped with the weak* topology. Define the $T$-relative entropy by $h_T :\mathcal{P}(X) \to \mathbb{R}$ by $$h_T(\nu) = -\int_X\log \frac{dT^{-1}_*\nu}{d\nu}(x)d\nu(x).$$

Is $h_T$ continuous? Is it upper semi-continuous?

Thanks!