I'm wondering how does one define finitely additive transition probabilities. Let $\mathcal X_1$ and $\mathcal X_2$ denote the Borel $\sigma$ algebra of two topological spaces $X_1$ and $X_2$. Suppose $\alpha_2:\mathcal X_2\to \mathbb R$ is a countably additive probability measure and $\alpha_1:\mathcal X_1\times X_2$ is such that $\alpha_1(\cdot|x_2)$ is a finitely additive probability measure for each $x_2$. What conditions can one impose on $\alpha_1$ so that the integrals

$$\int_{X_1}\int_{X_2} f(x_1,x_2)\, d\alpha_1(x_1|x_2)d\alpha_2(x_2) $$

makes sense for $f\in L^{\infty}(X_1\times X_2,\mathcal X_1\otimes \mathcal X_2)$?

Is it enough to require $\alpha(B|x_2)$ measurable in $x_2$ for each $B\in \mathcal X_1$?

When do we know that

$$\int_{X_1} f(x_1,x_2)d\alpha_1(x_1|x_2) $$

is measurable with respect to $x_2$?



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