This is a Corollary from Rene Schilling's Brownian Motion.
The proof below simply states that this is a corollary from the below equation (6.8) and the fact that $u(B_\tau)$ and $u(W_{\tau'} + B_\sigma)$ have the same distribution.
So we have $E[u(B_\tau) |\mathscr{F}_{\sigma+}](\omega) = E[u(W_{\tau'}+B_\sigma)|\mathscr{F}_{\sigma+}]$. And we need to show that this is equal to $E[u(W_{\tau'}+x)]|_{x=B_\sigma(\omega)}$ using (6.8).
However, $u$ here is a bounded measurable Borel function on $\mathbb{R}^d$, and $\Psi$ in (6.8) is a bounded $\mathscr{B}(\mathcal{C})/\mathscr{B}(\mathbb{R})$ measurable function. So I can't see how to define the $\Psi$ in (6.8) to get the first identity in (6.9).
How do we get the proof for this Corollary rigorously? I would greatly appreciate any help.
