What are the consistency criteria for Kolmogorov's existence theorem? I've been trying to better understand Kolmogorov's Existence Theorem, and I was reading a couple of references to get a better picture of what is going on. But then I got really confused, cause it seemed that each reference used a variation and I was not able to piece them together.
The main thing that confused me were the consistency conditions. If I read in Wikipedia and book like "Measure Theory and Probability Theory" (by Athreya), then one consistency condition is:
$$
\mu_{(t_1,...,t_{n+1})}(B_1 \times ...\times B_n \times \mathbb R) = 
\mu_{(t_1,...,t_n)}(B_1 \times ...\times B_n);
$$
And the second condition is permutation.
The issue is, in books like "Stochastic Calculus" (by Baldi) and some lecture notes,
the theorem is stated only with the first consistency condition, i.e. without permutation. Why is that so? I'm guessing there might be some "hidden" difference in the way the theorems are stated, but I could not figure out. Here is the theorem as stated by Baldi:
Consistency: Let $\pi = (t_1,...,t_n)$ and $\pi'=(t_1,...,t_{i-1},t_{i+1}, ..., t_n)$. Let $p_i : E^n \to E^{n-1}$ be the map defined as $(x_1,...,x_n)\mapsto (x_1,...,x_{i-1},x_{i+1},...x_n)$. Tjhe the image of $\mu_\pi$ through $p_i$ is equal to $\mu_{\pi'}$.
Theorem (Kolmogorov's Existence - Baldi) Let $E$ be a complete separable metric space,
with finite-dimensional distributions $(\mu_\pi)$ satisfying the consistency condition. Let $\Omega = E^T$ (the set of all paths from $T\to E$) and define $X_t(\omega) = \omega(t), \mathcal F_t( \sigma(X_s, s \leq t), \mathcal F = \sigma(X_t,t \in T)$. Then there exists a unique probability $P$ on $(\Omega, \mathcal F)$ with the $(\mu_\pi)$ family of finite-dimensional distributions.
Now, Baldi is doing things in quite a general context. For a simpler version, some lecture notes I'm using state Kolmogorov's theorem as the following:
Theorem (Kolmogorov's Existence - Notes) For each $n \geq 1$ consider a probability measure $P_n \in \mathbb R^n$ such that
$$
P_{n+1}(A\times \mathbb R) = P_n(A), \forall A \in \mathcal B(\mathbb R^n).
$$
Then there exists a unique probability measure $P$ in the space of infinite sequences $(\Omega, \mathcal F)$ such that $P(A \times \mathbb R \times ...) = P_n(A)$ for every $n$ and every $A \in \mathcal B(\mathbb R^n))$.
Finally, Durrett also expresses the theorem similarly without the permutation:
Theorem (Kolmogorov's Existence - Durrett) Suppose we are given probability measure on $(\mathbb R^n, \mathcal B(\mathbb R^n))$ that are consistent, that is
$$
\mu_{n+1}((a_1,b_1]\times ... \times (a_n,b_n] \times \mathbb R)
= \mu_n((a_1,b_1]\times...\times (a_n,b_n]).
$$
There there is a unique probability measure on $(\mathbb R^\mathbb N, \mathcal B(\mathbb R^\mathbb N))$.
Why there is no permutation condition in these versions?
 A: Billingsley to the rescue! The answer for this question can be found in Example 36.4 from Billingsley's "Probability and Measure", but it's a bit convoluted. I think I was finally able to piece everything together, so here is my take (please correct me if I got something wrong).
The problem addressed by Kolmogorov's Existence Theorem is actually two fold. First, we want to construct the probability measure $P$ in $(\Omega = E^T, \mathcal F = \mathcal B(E)^T)$, where $E^T$ is the set of functions going from $T$ to $E$. Now, not only we want to prove that there exists a probability measure in this space, but also that for the coordinate projection r.v $X_t(\omega) = \omega(t)$, we have that for every $t_1,...,t_n \in T$,
$$
P((X_{t_1},...,X_{t_n})\in A) = \mu_{(t_1,...,t_n)}(A),
$$
where $(\mu_t)_{t \in T}$ is the finite-dimensional distribution family.
Now, the issue that arises is that a finite-dimensional distribution family can be "inconsistent", i.e. there is no possible probability measure $P$ satisfying the condition above. Indeed, one can prove that
for a given probability measure $P$,
$$
P((X_{t_1},...,X_{t_n})\in A) = \mu_{(t_1,...,t_n)}(A) \implies (C1) \text{ and } (C2).
$$
where (C1) and (C2) are the "common" consistency conditions. Hence, if a family does not satisfy the consistency conditions, it cannot have an underlying probability measure as we stated.
So, what about the "single" condition in Durrett and Baldi? The answer is Example 36.4 from Billingsley. Note that (C1) and (C2) do not assume that $t_1<...<t_n$, but this condition appears in Baldi and implicitly in Durrett's (and the lecture notes), which define the finite-dimensional distributions to be restricted to ordered sets.
The fact is, for an ordered set $t_1<...<t_n$, we have by definition that
$\mu_{(t_1,...,t_n)}$ is the distribution of $(X_{t_1},...,X_{t_n})$.
And what if a set $s_1,...,s_n$ is a permutation of $t_1,...,t_n$,i.e.
if the set is not ordered, then what is $\mu_(s_1,...,s_n)$?
This is actually undefined! Hence, we can just define it as the distribution of $(X_{s_1},...,X_{s_n})$. In other words, we are assuming "implicitly" the permutation condition when we only provide the ordered set of finite-dimensional distributions. With this, we can then prove that Baldi's condition is the same as (C1) and (C2). The same for Durrett and the lecture notes.
