Possibly! Extending factored sets to continuous variables is an active area of research.
Scott Garrabrant has found 3 different ways to extend the orthogonality and time definitions to infinite sets, and it is not clear which one captures most of what we want to talk about.
I suspect that the fundamental theorem can be extended to finite-dimensional factored sets (i.e., factored sets where |B| is finite), but it can not be extended to arbitrary-dimension factored sets
If his suspicion is right, that means we can use factored sets to model continuous variables, but not model a continuous set of variables (e.g. we could model the position of a point in space as a continuous random variable, but couldn’t model a curve consisting of uncountably many points)
Possibly! Extending factored sets to continuous variables is an active area of research.
Scott Garrabrant has found 3 different ways to extend the orthogonality and time definitions to infinite sets, and it is not clear which one captures most of what we want to talk about.
About his central result, Scott writes:
If his suspicion is right, that means we can use factored sets to model continuous variables, but not model a continuous set of variables (e.g. we could model the position of a point in space as a continuous random variable, but couldn’t model a curve consisting of uncountably many points)
The Countably Factored Spaces post also seems very relevant. (I have only skimmed it)