I am reading Schilling's textbook, 'Brownian Motion: An Introduction to Stochastic Processes (2nd ed.)'.
I couldn't find way to prove (6.5a) in page 61.
By the textbook, for a given $ \mathbb{R}^d $ -valued filtered stochastic process $ (X_t, \mathcal{F}_t)_{t \geq 0} $, the process is called homogeneous Markov process if the following holds.
For all bounded $ \mathfrak{B}(\mathbb{R}^d) / \mathfrak{B}(\mathbb{R})$-measurable map $u$ and $t, s \geq 0$, there exists a measurable function $g_{u, t}: \mathbb{R}^d \to \mathbb{R}$ which only depends on $u$ and $t$ so that $$ \mathbb{E}\left[ u(X_{t+s}) | \mathcal{F}_s \right] = \mathbb{E} \left[ u(X_{t+s}) | X_s \right] = \mathbb{E}^{X_s} \left[ u(X_t) \right] $$ where $ \mathbb{E}^x [u(X_t)] := g_{u, t}(x)$.
Book claims that by using standard measure theory technique, one can prove that following (6.5a) holds.$$ \mathbb{E} \left[ \Psi(X_{\cdot+s}) | \mathcal{F}_s \right] = \mathbb{E} \left[ \Psi(X_{\cdot+s}) | X_s \right] = \mathbb{E}^{X_s} \left[ \Psi(X_\cdot) \right] $$ for all bounded measurable functionals $\Psi: \mathcal{C}([0, \infty)) \to \mathbb{R}$.
I think in this book, Borel $\sigma$-algebra of $\mathcal{C}([0, \infty))$ should be the one induced by the topology of locally uniform convergence (with the metric $\rho(f, g) = \sum_{n=1}^\infty \left( 1 \wedge \sup_{0 \leq t \leq n} |f(t) - g(t)| \right) 2^{-n}$.)
So I tried to proceed by first proving it when $\Psi$ is an indicator function, then a simple function. Finally proving it for general cases by passing to the limit. But I got stuck at the first step.
My questions are (1) Am I thinking in the right way about the $\sigma$-algebra of $\mathcal{C}([0, \infty))$ here? (2) Any hints or ideas to prove (6.5a) in the case of an indicator function?
The first step is to look at functionals $\Psi$ that only depend on finitely many values: $\Psi(f)$ should just depend on $f(x_1),\ldots,f(x_n)$ for some finite set $\{x_1, \ldots,x_n\}$. These can be handled by induction on $n$. Then show that linear combinations of such functionals are dense in the space of bounded measurable functionals. This follows from the monotone class theorem or the Dynkin $\pi-\lambda$ theorem.