Is $x \mapsto P_x(A)$ measurable for a measure $P_x$ determined by transition kernels $(\delta_x,P_i)_{i \ge 0}$

Question

Given a sequence $P_i$ of transition kernels from $(E,\mathcal B(E)$ to $(E,\mathcal B(E))$ and $\delta_x$ the Dirac delta measure for the point $x\in E$, it follows from the Ionescu-Tulcea theorem that for every $x$ there exists a unique measure $P_x$ which is consistent with the $n $ dimensional distributions $B \mapsto \int \delta_x(d x_o) \int...\int P_n(x_{n-1},d x_n)1_B(x_0,...,x_n))$

My question is: Given an abitrary collection of measures $P_x$ on some measurable space $(E, \mathcal E )$ and some $A \in \mathcal E $ would it be true that the map $x \mapsto P_x(A)$ is measurable. If not is it then true in the present situation?

Thanks in advance!

Answer 1

I believe I may answer the second question of my own post! If we treat $\delta_x $ as a kernel given by $(x,A) \mapsto \delta_x (A), \ (x,A) \in (E, \mathcal E)$ the result follows as proved by saz here.