I have a problem with the proof of Theorem 5.1 in chapter 4 of the book "Stochastic Differential Equations and Diffusion Processes" by Ikeda, Watanabe.
First, it makes the following definitions:
- Let $S$ to be a topological space and defines $S'=S\cup \{\Delta\}$, where either $\Delta$ is an isolated point or if $S$ is locally compact is a point at infinity (one point compactification.)
- Let $\bar{W}(S)$ as the set of all functions $w:[0,\infty)\to S'$ such that there exists $0\leq \zeta(w)\leq \infty$, where $w(t)\in S$ for all $t\in[0,\zeta(w))$, and $t\in[0,\zeta(w))\mapsto w(t)$ is continuous. Finally, $w(t)=\Delta$ for all $t\geq \zeta(w)$.
- Let $\{P_x,x\in S\}$ be a system of diffusion measures on $(\bar{W}(S),\mathscr B(\bar{W}(S)))$.
In the proof of the theorem it defines $\tilde{P}^w(A)=P_x(\theta^{-1}_\sigma(A)|\mathscr F_\sigma(\bar{W}(S)))$, where $A\in\mathscr B(\bar{W}(S)))$, $\sigma$ is a bounded stopping time with the sigma algebra $\mathscr F_\sigma(\bar{W}(S)))$, and $\theta$ is the shift.
Now my problems:
It argues without explanation that $\tilde{P}^w$ is a regular conditional probability given $\mathscr F_\sigma(\bar{W}(S))$. I cannot see why regular conditional probability exists on $\bar{W}(S)$. Given the very general definition of $S$ and $S'$ I do not think $\bar{W}(S)$ is necessarily a standard Borel (nice) space. So how do we know such regular conditional probability exists?
I think the definition of $\tilde{P}^w(A)$ should be $\tilde{P}^w(A)=P_x(\theta^{-1}_\sigma(A)|\mathscr F_\sigma(\bar{W}(S)))(w)$, since $P_x(\theta^{-1}_\sigma(A)|\mathscr F_\sigma(\bar{W}(S)))$ is not a measure but a random measure or a kernel.
I appreciate if anyone could clarify my above confusions.