Is there an extension of maximum entropy probability distributions for function spaces?

For $\mathbb{R}^n$ and discrete spaces, there is much literature about this problem under names such as "non-informative priors", "maximum entropy distributions", "Jeffrey's priors", and the like.

There is an extension to locally compact topological groups, where the Haar measure $U$ takes the place of the Lebesgue measure, and one looks for measures $P$ minimizing the information divergence, $$D(P||U):= \begin{cases} \int log \frac{dP}{dU} dP, & \text{ if } P\ll U; \\\ \infty, & \text{else.}\end{cases}$$

However, I've found little about this in the infinite dimensional setting. Can the concept of maximum entropy priors be generalized to (some class of) function spaces, or is the idea of entropy fundamentally incompatible with spaces that are not locally compact?

Notes,

- This is a repost from this statistics stackexchange thread which got no answers.
- There is an infinite dimensional generalization of the "min- entropy" $H_\infty$, though this is a different concept from the "minimum entropy" $\text{min } H$.