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?



Necessary and sufficient conditions for Kolmogorov's Extension Theorem
Gnedenko and Kolgomorov defined perfect measures for this purpose. Perfectness is a sufficient condition: If all the marginal measures (projections to $X\_n$) are perfect then the resulting finitely additive measure on the product is countably additive. In this form perfectness is not necessary. But it is necessary in slightly modified formulations. For example, a probability measure $\mu$ is perfect if and only if for every probability measure $\nu$ every product of one copy of $\nu$ and any number of copies of $\mu$ is countably additive. This can be deduced from Ryll-Nardzewski's paper "On quasi-compact measures" in Fund. Math. 40 (1953).
See also section 454 in Fremlin's Measure Theory.


If the condition you are looking for is on the measurable spaces rather than the measures on them, the corresponding condition is "every countably additive measure on the space is perfect".


