I don't understand the following claim from my book:
Let $(B_t)$ be a standard Brownian motion.
Let $u:\Omega \rightarrow \mathbb{R}$ be a continuous function, where $\Omega$ is a domain and $B(x, \delta) \subset \subset \Omega$. Also, let $$ \tau = \inf \{ t>0 : B_t + x \in \partial B(x, \delta) \} \quad \text{ and } \quad \tau(\partial \Omega) = \inf \{ t>0 : B_t + x \in \partial \Omega \}. $$ Then, it claims that, by strong Markov property, $$\mathbb{E} \bigg( u (B_{\tau (\partial \Omega)} + x ) \bigg| \mathcal{F}_\tau \bigg) = \mathbb {E} \bigg( u \big( B_{\tau (\partial \Omega)} +y \big) \bigg) \bigg|_{y=B_{\tau}+x}. $$
But I get
\begin{eqnarray} \mathbb{E} \bigg( u (B_{\tau (\partial \Omega)} + x ) \bigg| \mathcal{F}_\tau \bigg) & = & \mathbb{E} \bigg( u \big( B_{\tau + (\tau (\partial \Omega) - \tau \big)} - B_{\tau} + B_{\tau} + x ) \bigg| \mathcal{F}_\tau \bigg) \\ & = & \mathbb {E} \bigg( u \big( B_{ \{\tau (\partial \Omega) - \tau \} } +y \big) \bigg) \bigg|_{y=B_{\tau}+x}, \end{eqnarray} since $u$ is a Borel function, $\{B_{\tau+t}- B_\tau \}_{t \geq 0}$ is a Brownian motion independent of $\mathcal{F}_{\tau}$ and $B_{\tau}+x$ is $\mathcal{F}_{\tau}$-measurable.