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://gymarkiv.sdu.dk/MFM/kdvs/mfm%2020-29/mfm-25-4.pdf
  [2]: https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/comparison-of-methods-for-constructing-probability-measures-on-infinite-product-spaces/DCF12E254694F751A0A116C957181A20