I'm exploring the possibility to apply information theory on an uncountably infinite-dimensional scenario. I found the concept of generalized entropy for continuous random variables defined on finite-dimensional Euclidean space. I wonder whether there are similar concepts for uncountably infinite-dimensional cases. For instance, if I have a distribution on $C([0, 1])$, which is the space of continuous function on $[0, 1]$, then how can I quantify the entropy of this distribution?