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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?

Any conditions enabling this existence or references to litterature about this is much appreciated. Thanks.

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It turns out that this is easily proved using a Dynkin-class argument. See here in the section on measure theory.

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