[KET](http://en.wikipedia.org/wiki/Kolmogorov_extension_theorem) 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?




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I also asked a related question [here](https://math.stackexchange.com/questions/267568/kolmogorov-extension-theorem-vs-caratheodory-extension-theorem) but didn't get any answer.



Topological conditions of Kolmogorov Extension Theorem
The KET fails for general measurable spaces, the classical example can be found in a [paper](http://gymarkiv.sdu.dk/MFM/kdvs/mfm%2020-29/mfm-25-4.pdf) by Andersen and Jessen. Topological assumptions are necessary so that the resulting measure is not only finitely additive but countably additive. There exists a quasi-topological condition of measure spaces, *perfectness*, that is sufficient. A probability space $(\Omega,\sigma,\mu)$ is perfect if for every random variable $f:\Omega\to\mathbb{R}$, there exists a Borel set $B\subseteq f(\Omega)$ with measure one under the distribution $\mu\circ f^{-1}$. A proof of KET under the assumption that the marginal measures are perfect due to Lamb is given [here](https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/comparison-of-methods-for-constructing-probability-measures-on-infinite-product-spaces/DCF12E254694F751A0A116C957181A20). The strategy of the proof is to employ an existence result for regular conditional probability spaces and the construct the proces for them using the Ionescu-Tulcea theorem.


