@PietroMajer: so we can't define some/most interesting measures on power sets while having the countably additive $\implies$ $\sigma$-fields $\implies$ measurability?

@PietroMajer: indeed, and his research is also cited in the section in Dubins and Savage that I've mentioned in OP. So do you mean that the requirement of measurability is mostly a technical drawback of a $\sigma$-additive setting?

@GeraldEdgar: I see, could you suggest any reference to this fact? Perhaps, a standard textbook treats it - but unfortunately I am not familiar with such statement.

@GeraldEdgar: thanks, that could be a nice idea, though I don't see why an extreme has necessary to be some deterministic map.

@GeorgeLowther: thanks for the comments. As far as I got from the proof there, if one shows that Lemma 1 holds for analytical $S$ (= there exists a Borel $\tilde S\subset S$ with $\mu_x$-positive closed $x$-sections), then the existence of a Borel graph contained in $S$ is immediate. If that is what you meant, could you suggest how to show existence of $\tilde S$ for analytical $S$? I also don't quite understand, why do they apply Lemma 1 to the completion of $Y$, and why do such $F_k$ exist - we can assume that when $Y$ is completed, it is totally bounded?

I would be happy if anybody could at least help me simplifying the setting. I guess, that for the solution of the problem it does not matter which are the spaces $X^n$ and $\mathscr P(X)$ and one can go just for some continuous surjective map in place of $\pi_X$. However, I don't know how to simplify/make neater constraints related to $K$.

Can you say, what does local harmonicity means in terms of Markov Chains?