Given a transition kernel $p $ on $\mathbb R$: we may define a consistent collection of finite dimensional distribution by letting $X_i(\omega) = \omega$ and writing
$$\mathbb P_n (X_0 \in B_0, \dots, X_n \in B_n ) = \int_{B_0 } \mu(dx_0) \int_{B_1 } p(x_0, dx_1) \ \dots \int_{B_n } p(x_{n-1 } ,d_xn) $$
It is known [See Ionescu-Tulcea theorem] that this may be extended to a probability measure $\mathbb P_\mu $ on $\mathbb R^{\mathbb{N}} $.
I would like to show that given any transition kernel $p$, a sequence $(X_n)$ and a probability measure $\mathbb P $ that satisfies the equation above, that
$$\mathbb P[X_{n+1 }\in B|\mathcal{F}_n]=p(X_n,B)$$
This is what I have tried to gather:
For the claim to be true we should show that for any $A \in \sigma(X_0, \dots, X_n )$ and $B \in \mathcal{B}(\mathbb{R})$
$$\int _A 1_{\{X_{n+1 } \in B \} } d \mathbb P = \int_A p(X_n, B) d \mathbb P$$
[I am a little bit unsure here which measure we should integrate with respect to as I have seen some sources write this integral with respect to $\mathbb P_\mu$ but that does not make sense to me since $p(X_n, B)$ is a function from $\Omega$ if $X_n: \Omega \to \mathbb{R}$.]
If it would be sufficient to consider $A $ of the form $\{X_0 \in B_0, \dots, X_n \in B_n \} $ (which I believe it is since I think those sets generate $\mathcal{F}_n$?) then we would have
\begin{multline} \int_A 1_{\{X_{n+1 } \in B \} } d \mathbb P = \mathbb P(X_0 \in B_0, \dots, X_n \in B_n, X_{n+1 } \in B) \\ = \int_{B_0} \mu(d x_0) \int_{B_1} p(x_0, dx_1) \dots \int_{B_n} p(x_{n-1 }, dx_n) p(x_n, B) \end{multline}
If we could in fact show that
$$\int_{B_0} \mu(d x_0) \int_{B_1} p(x_0, dx_1) \dots \int_{B_n} p(x_{n-1 }, dx_n) f(x_n) = \int_A f(X_n(\omega)) d \mathbb P $$
for bounded measurable $f$, then (1) would follow.
To prove the equality for bounded measurable $f$ it would be sufficient to prove it for an indicator function and then use the customary approximation arguments. So I tried this for $f = 1_D $ and got
\begin{multline*} \int_A 1_D d \mathbb P = \mathbb P(A \cap D) \\ = \int_{B_0 \cap D} \mu(dx_0) \int_{B_1 \cap D } p(x_0, dx_1) \dots \int_{B_n \cap D } p(x_{n-1 } dx_n) \end{multline*}
Does
\begin{multline*} \int_{B_0} \mu(d x_0) \int_{B_1} p(x_0, dx_1) \dots \int_{B_n} p(x_{n-1 }, dx_n) 1_D(x_n) \\ = \int_{B_0 \cap D} \mu(dx_0) \int_{B_1 \cap D } p(x_0, dx_1) \dots \int_{B_n \cap D } p(x_{n-1 } dx_n) \end{multline*}
?
Thus my questions are 1) the last equation. 2) If it is sufficient to consider $A$ of the form $\{X_0 \in B_0, \dots, X_n \in B_n \}$? and 3) If I got it right integrating w.r.t. $\mathbb P$ - the measure on the space on which $(X_n) $ is defined?
Most grateful for any help provided!