# Ionescu Tulcea kernel

Let $$(\mathcal{X}_i, \mathcal{A}_i)$$ be a measurable space, $$\kappa_i : \mathcal{A}_i \times \prod_{j=1}^i \mathcal{X}_j \to [0,1]$$ be a Markov kernel for $$i \in \mathbb{N}$$ and $$\mu_1$$ be a probability measure on $$\mathcal{A}_1$$. Ionescu-Tulcea says that for each $$i \in \mathbb{N} \cup \{\infty\}$$ there exists a unique probability measure $$\mu_i$$ on $$\bigotimes_{j=1}^i \mathcal{A}_j$$ with $$\mu_i(A_1 \times \dots \times A_i) = \mu_\infty \left(A_1 \times \dots \times A_i \times \prod_{j=i+1}^\infty \mathcal{X}_j \right)$$ Edit: With some encouragement from a friend who was so kind to review this question I think I can simplify my question a bit:

Notice that $$\kappa_1(\cdot \mid x)$$ defines a measure for each $$x \in \mathcal{X}_1$$. Let $$\mu_x$$ be the Ionescu-Tulcea measure generated by $$\kappa_1(\cdot \mid x)$$ and $$\kappa_2, \kappa_3, \dots$$. Let $$\tau(A \mid x) = \mu_x(A)$$. Is $$\tau$$ a (Markov) kernel?