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

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?

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.