KET is often used to construct stochastic processes in continuous time when the state space is $\Bbb R^d$. As far as I am familiar with its proof, it uses standard monotonic class-like arguments together with Caratheodory Extension Theorem. Neither of the two latter theorems requires any topological conditions. Moreover, I have not been able to find where KET uses that the space is $\Bbb R^d$ rather than just some measurable space.
Q: is it true that KET holds for general measurable spaces, and if not - where is the topology of $\Bbb R^d$ crucial in the proof? Also, maybe any counterexample (for exsitence or uniqueness) is known?
I also asked a related question here but didn't get any answer.