Let $(X_n,\mathcal{X}_n)$, $n=1,2,\ldots$ be measurable spaces. Define $Y_n = \prod_{k=1}^n X_k$ and let $\mathcal{Y}_n$ be the corresponding product $\sigma$-algebra. Similarly let $Y=\prod_{k=1}^\infty X_k$ and $\mathcal{Y}$ the corresponding $\sigma$-algebra.
Let $\mu_n$ be probability measures on $(Y_n,\mathcal{Y}_n)$ which are consistent in the sense that the projection of $\mu_{n+1}$ on $Y_n$ is $\mu_n$.
The Kolmogorov Extension Theorem (KET) states that under additional conditions on the spaces $X_n$ there exists a probability measure $\mu$ on $Y$ such that its projection on each $Y_n$ is $\mu_n$.
I have come across different versions of the KET which impose different conditions on the $X_n$. My question is whether there exist necessary and sufficient conditions which characterize spaces where consistent probability measures on finite product spaces can always be extended to the infinite product?
The extension problem can be generalized by looking at a fixed set $Y$ and considering an increasing sequence of $\sigma$-algebras $\mathcal{Y}_n$ and measures $\mu_n$ on $(Y,\mathcal{Y}_n)$ such that $\mu_{n+1}$ and $\mu_n$ agree on $\mathcal{Y}_n$. The extension problem then is to find a probability measure on $\sigma(\cup_{k=1}^\infty \mathcal{Y}_k)$ which agrees with $\mu_n$ on each of the $\mathcal{Y}_n$? Once again, are there necessary and sufficient conditions for this problem?