The KET fails for general measurable spaces, the classical example can be found in a [paper][1] 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][2]. 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.


  [1]: http://www.sdu.dk/media/bibpdf/Bind%252020-29%255CBind%255Cmfm-25-4.pdf
  [2]: http://cms.math.ca/10.4153/CMB-1987-040-x