What is the Kolmogorov Extension Theorem good for?
The Kolmogorov Extension Theorem says, essentially, that one can get a process on $\mathbb{R}^T$ for $T$ being an arbitrary, non-empty index set, by specifying all finite dimensional distributions in a "consistent" way. My favorite formulation of the consistency condition can be found here. Now for the case in which $T$ is countable, this has already be shown by P. J. Daniell (see for example here or here). So I would like to know what the extension to uncountable index sets brings. Events like "sample paths are continuous" are not in the $\sigma$-algebra. In a rather critical paper on Kolmogrov's work on the foundation of probability, Shafer and Vovk write about the extension to uncountable index sets: "This greater generality is merely formal, in two senses: it involves no additional mathematical complications and it has no practical use." My impression is that this sentiment is not universally shared, so I would like to know:
How is the Kolmogorov Extension Theorem applied in the construction of stochastic processes in continuous time? Especially, how are the constructed probabilities transferred to richer measurable spaces?
Assume that you have a set of finite-dimensional distributions $(\mu_S)_{S\in A}$, where $A$ is the set of finite subsets of, say, $\mathbb{R}$, and assume that you would like to argue for the existence of a stochastic process $X$ with càdlàg (right-continuous with left limits) paths such that the family of finite-dimensional distributions of $X$ is $(\mu_S)_{S\in A}$. Kolmogorov's extension theorem allows you to split this problem into two parts:
Establishing the existence of a measure on $(\mathbb{R}^{\mathbb{R}_+},\mathbb{B}^{\mathbb{R}_+})$ with the appropriate finite-dimensional distributions (here, the extension theorem is invoked).
Using the measure constructed above, argue for the existence of a measure on $D[0,\infty)$ - the space of functions from $\mathbb{R}_+$ to $\mathbb{R}$ which are right-continuous with left limits - with the same finite-dimensional distributions.
One example of where this is a viable proof technique is in the theory of continuous-time Markov processes on general state spaces. For convenience, consider a complete, separable metric space (E,d) endowed with its Borel $\sigma$-algebra $\mathbb{E}$. Assume given a family of probability measures $(P(x,\cdot))_{x\in E}$, where $P(x,\cdot)$ is a probability measure on $(E,\mathbb{E})$. We wish to argue for the existence of a càdlàg continuous-time Markov process with values in $E$ and $(P(x,\cdot))_{x\in E}$ as its transition probabilities.
The following argument is given in Rogers & Williams: "Diffusions, Martingales and Markov Processes", Volume 1, Section III.7, and uses the two steps outlined above. First, the Kolmogorov extension theorem is invoked to obtain a measure $P$ on $(E^{\mathbb{R}_+},\mathbb{E}^{\mathbb{R}_+})$ with the desired finite-dimensional distributions. Letting $X$ be the identity on $E^{\mathbb{R}_+}$, $X$ is then a "non-regularized" process with the desired finite-dimensional distributions. Afterwards, in the case where the family of transition probabilities satisfy certain regularity criteria, a supermartingale argument is applied to obtain a càdlàg version of $X$. This supermartingale argument could not have been immediately applied without the existence of the measure $P$ on $(E^{\mathbb{R}_+},\mathbb{E}^{\mathbb{R}_+})$: Without this measure, there would be no candidate for a common probability space on which to define the supermartingales applied in the regularity proof. Thus, it is not obvious how to obtain the same existence result without the Kolmogorov extension theorem.