As far as I know, a class of spaces for which the Kolmogorov theorem works and which is closed under countable products, are the spaces of complete separable metric spaces which are also called Polish spaces.
1) This is the most extended version of the theorem? And Polish space is strictly the minimum condition? What is the minimum condition for any kind of uncountable index set?
2) The components of the infinite product used to be $(\Omega_n,\mathcal F_n,\mathbb P_n)$ in the literature, where $\Omega_n$ is a set, $\mathcal F_n$ is a sigma-algebra and $\mathbb P_n$ is a probability measure for all $n$ in the index set. However in the definition of Polish space there is metric and topology, it is not clear for me, that the metric and topology has to be related or not, only there existence is important or has to be defined and used, for example the measure has to be related somehow to the metric or the topology or not? I cannot find clear description in any books or lecture notes.
3) $\Omega_n$ and/or $\mathcal F_n$ and/or $\mathbb P_n$ has to be the same or can be different for all uncountable or infinitely countable components in the product? And in case of Polish spaces the metric and topology has to be the same?