I'm reading a book 'Introduction to theory of Random process' written by N. V. Krylov.
The author prove the following theorem via characteristic functions (p.53)
Theorem. Let $(w_t,\mathcal{F}_t)$ be a Wiener process. Fix $t, h_1,\dots,h_n\geq 0$. Then the vector $(w_{t+h_1} - w_t,\dots,w_{t+h_n} - w_t)$ and the $\sigma$-field $\mathcal{F}_t$ are independent. Furthermore, $w_{t+s}-w_t$, $s\geq 0$ is a Wiener process.
By some reduction, it suffices to show $\eta_n =(w_{t+h_1}-w_{t+h_0},\dots,w_{t+h_n} -w_t)$ and the $\sigma$-field $\mathcal{F}_t$ are independent.
For any $A\in \mathcal{F}_t$, $\lambda\in\mathbb{R}^n$, following some inductive argument in the book, we have $$ \mathbb{E} I_A \exp(i\lambda\cdot \eta_n) =\mathbb{P}(A)\mathbb{E} \exp (i\lambda\cdot \eta_n)$$ holds.
Here is my question. The author claim that `It follows from the theory of characteristic function that for every Borel bounded $g$, we have $$ \mathbb{E} I_A g(\eta_n) = \mathbb{P} (A) \mathbb{E} g(\eta_n). $$
How can I prove it? I tried to use trigonometric approximation by using Weierstrass theorem and using Egorov's theorem, but I failed to use it due to some technical problem.
It seems to use inversion formula, but I cannot handle the integral part to desired form.
I didn't find any references on this fact.
Thanks in advance.