When wishing to prove existence of Brownian motion, most authors take the route of defining the desired finite dimensional distributions, showing the family of defined finite dimensional distributions is "consistent" and then applying the Daniell-Kolmogorov extension theorem to claim there exists a probability measure on the measurable space $(\mathbb{R}^{[0,\infty)},\mathcal{F})$ where $\mathbb{R}^{[0,\infty)}$ is the space of real valued functions on $[0,\infty)$ and $\mathcal{F}$ is the sigma algebra generated by the algebra of finite dimensional cylinder sets in $\mathbb{R}^{[0,\infty)}$ . I have found some inconsistencies between authors on how they define "consistent" and I feel uneasy about how the extension theorem is applied in certain situations.

First I will make clear what I mean by a family of finite dimensional distributions. Let $T$ set of finite sequences of the form $t=(t_1,t_2,\ldots,t_k)$ of distinct nonnegative numbers whose length $k$ takes values in the set of positive integers. Then suppose for each $t$ of length $k$ we have a probability measure $\mathbb{P}_t$ on $(\mathbb{R}^k,\mathcal{B}(\mathbb{R}^k))$. We then call $(\mathbb{P}_t)$ for all $t\in T$ a family of finite dimensional distributions.

Karatzas and Shreve say such a family is consistent if the following two conditions hold,

(i.) If $s=(t_{i_1},t_{i_2},\ldots,t_{i_k})$ is a permutation of $t=(t_1,t_2,\ldots,t_k)$, then for any collection of Borel sets $(B_i)_{i=1}^k$ we have

$$\mathbb{P}_{t}(B_1 \times B_2 \times \ldots \times B_k)=\mathbb{P}_{s}(B_{i_1} \times B_{i_2} \times \ldots \times B_{i_k})$$

(ii.) If $t=(t_1,t_2,\ldots,t_k)$ with $k \geq 1$, $s=(t_1,t_2,\ldots,t_{k-1})$ and $\{B_i\}_{i=1}^{k-1}$ is a collection of Borel sets then,

$$ \mathbb{P}_{t}(B_1 \times \ldots \times B_{k-1} \times \mathbb{R}) = \mathbb{P}_{s}(B_{1} \times B_{2} \times \ldots \times B_{{k-1}}).$$

Now in the book by Kallianpur and Sundar they only insist on the second condition and state that the set of indices $t=(t_1,\ldots,t_k)$ must be strictly ordered. Both authors then apply the extension theorem abiding by their definition of a consistent family of finite dimensional distributions. Now for simplicity let $k=2$ and define the finite dimensional distribution of Brownian motion for any arbitrary $x \in \mathbb{R}$ and $t=(t_1,t_2)$ where $0\leq t_1 \leq t_2$ as,

$$ \mathbb{P}_{t}(B_1 \times B_2) = \int_{B_1 \times B_2} \frac{\exp(-\frac{|x-y|^2}{2t_1})}{\sqrt{2 \pi t_1}} \cdot \frac{\exp(-\frac{|y-z|^2}{2(t_2-t_1)})}{\sqrt{2 \pi(t_2- t_1)}} dydz $$.

I believe the first condition will not be satisfied as you cannot permute $t_1$ and $t_2$. You need the strict ordering. My questions is then am I missing something trivial, as pretty much all famous books use the conditions as in Karatzas and Shreve.

The second book is *Stochastic analysis and diffusion processes* by Gopinath Kallianpur and Padmanabhan Sundar.