I'm trying to understand the problem arising when using Kolmogorov's extension theorem to prove the existence of the Dirichlet process on an arbitrary measurable space $\left(\mathcal{X},\mathcal{A}\right)$.

Recalling the definition from Ferguson: A $\left[0,1\right]$-valued process $\left(\mu_{A}\right)_{A\in\mathcal{A}}$ is a Dirichlet process with parameter $\alpha\in\mathcal{M}_{1}\left(\mathcal{X}\right)$ if for any finite measurable partition $A_{1},\ldots,A_{k}$ of $\mathcal{X}$:

$$ \left(\mu_{A_{1}},\ldots,\mu_{A_{k}}\right)\sim\mbox{Dir}\left(\alpha H\left(A_{1}\right),\ldots,\alpha H\left(A_{k}\right)\right) $$

where $\mbox{Dir}$ denotes the Dirichlet distribution.

Since this family of marginals is consistent (follows from the “aggregation property” of the Dirichlet), Kolmogorov's extension theorem tells me that there's a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ supporting this process and that a can choose $\Omega=\left[0,1\right]^{\mathcal{A}}$, $\mathcal{F}=\mathcal{B}\left(\left[0,1\right]\right)^{\otimes\mathcal{A}}$ and $\mu_{A}\left(\omega\right)=\omega_{A}$.

My understanding is that Kolmogorov's does not give us any properties of $\mu$ besides its existence, and that we need to show that it almost surely is a probability measure. The tricky point seems to be the $\sigma$-additivity: For any countable collection of measurable sets $\mathcal{D}\subset\mathcal{A}$ we need

$$\mu\left(\bigcup_{A\in\mathcal{A}}A\right)=\sum_{A\in\mathcal{A}}\mu\left(A\right)\mbox{a.s.} $$

and if the $\sigma$-algebra $\mathcal{A}$ does not have a countable generator, then there are uncountably many countable collections $\mathcal{D}$.

**Now my questions are:**

(Probably trivial:) Why does this equality even hold for a single countable collection? Doesn't this require me to know countable marginals where I only know finite ones?

What is exactly the problem with uncountability? Is it that null sets sum up, so that potentially $\mathbb{P}\left[\mu\mbox{ is }\sigma\mbox{-additive}\right]<1$ even if the above equation holds for every collection of measurable sets? Or is the event $\left\{ \mu\mbox{ is }\sigma\mbox{-additive}\right\}$ not even measurable w.r.t the product algebra $\mathcal{F}$?

What gets easier by assuming that $\mathcal{X}$ is polish? Does every Borel-$\sigma$-algebra on a polish space have a countable generator?

My references are:

- Ferguson (1973), "A Bayesian analysis of some nonparametric problems"
- Sethuraman (1991), "A constructive definition of Dirichlet priors"