Let $(X, | \cdot |)$ be a Banach space.
I am interested in whether one can extend the definition of the Kullback-Leibler divergence $$ \text{KL}(\mu \ \Vert \ \nu) := \int_{\Omega} \ln\left(\frac{\text{d}\mu}{\text{d}\nu}\right) \; \text{d}\mu - \mu(\Omega) + \nu(\Omega), $$ which is defined for measures $\mu$, $\nu$ on $\Sigma$, where $\Sigma$ is a $\sigma$-algebra of some set $\Omega$, with $\mu \ll \nu$ (that is, $\mu$ is absolutely continuous with respect to $\nu$) so that the Radon-Nikodym derivative $\frac{\text{d}\mu}{\text{d}\nu} \in L^1(\Omega, \mu)$ exists, to $X$-valued measures $\mu, \nu$ of bounded semi-variation.
One problem clearly is to give meaning to the Radon-Nikodym derivative $\frac{\text{d}\mu}{\text{d}\nu}$ if both measures are vector-valued.
A standard definition (cf. Diestel, Uhl: Vector measures, Def. 3 in Chp. 3, p. 61) is:
Definition 1 (Radon-Nikodym property). Let $(\Omega, \Sigma, \mu)$ be a finite measure space, that is, $\mu \colon \Sigma \to [0, \infty)$ be a countably additive measure and $\Sigma$ a $\sigma$-algebra over a set $\Omega$ with $\mu(\Omega) < \infty$. A Banach space $X$ has the Radon-Nikodym property with respect to $(\Omega, \Sigma, \mu)$ if for each $\mu$-continuous vector measure $G \colon \Sigma \to X$ of bounded variation there exists a $g \in L_1(\mu, X)$ (the Radon-Nikodym derivative of $G$ with respect to $\mu$) such that $$ G(E) = \int_{E} g \; \text{d}\mu $$ for all $E \in \Sigma$. Lastly, $X$ has the Radon-Nikodym property if it has the Radon-Nikodym property for all $(\Omega, \Sigma, \mu)$.
Here, $L_1(\mu, X) := L_1(\Omega, \Sigma, \mu; X)$ is the space of (equivalence classes of) Bochner-integrable functions $\Omega \to X$ and $G$ is $\mu$-continuous if $\lim_{\mu(E) \to 0} G(E) = 0$ (p. 49 and 10 in Diestel, Uhl).
My question is:
What happens if we instead let $\mu \colon \Sigma \to Y$ be a vector-valued measure with values in a Banach space $Y$ such that there exists a bilinear map $X \times Y \to Z$, $(x, y) \mapsto x y$, where $Z$ is a Banach space such that there exists a constant $K > 0$ with $| x y | \le K | x | | y |$ for all $x \in X$ and $y \in Y$? Is this extended definition of the Radon-Nikodym property meaningful, e.g. would it still be true that e.g. reflexive spaces have the Radon-Nikodym property?
This bounded-bilinear-form condition is sufficient for the integral $\int_{E} g(s) \; \text{d}\mu(s) \in Z$ to be well-defined for $g \colon \Omega \to X$, $\mu \colon \Sigma \to Y$ and $E \in \Sigma$ (cf. Bartle: A general bilinear vector integral, 1956).
Note: There already is a generalisation of the KL divergence to measures taking values in the cone of real symmetric positive semidefinite matrices in Quantum Optimal Transport for Tensor Field Processing by Péyre et al. in section 1.3.
