The proof is written allowing for a state space that is an arbitrary measurable space, but I am interested in the case of real valued stochastic processes.
I am working through the proof of the theorem, where we are in the process of trying to show that finite dimensional distributions defined on $\mathbb{R}^F$ can be extended onto $\mathbb{R}^T$ where $T$ is an arbitrary index set and $F\subset T$ with $F$ finite.
On page 441: "First an easy proof will be given if the $(S_t, \mathcal{B}_t)$ are all standard measurable spaces, such as $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. So we can assume that $S_t$ are all compact metric spaces with Borel $\sigma$-algebras. Thus $S_T$ is compact by Tychonoff's theorem (2.2.8)."
Surely this is an error because $\mathbb{R}$ is not compact, hence we cannot invoke Tychonoff's theorem that $R^T$ is compact.
Also it is not even clear to me how the definition given a standard measurable space given on page 440 can be correct because a counterexample would be again be $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$.
Am I right in saying we just need to replace the word "compact" with the word "complete" and ignore the part where Tychonoff's theorem is used and instead try and show that the product of complete metric spaces is complete?
Thanks for any help.