Conditioning in Two Steps: Composing a Conditional Kernel Given a Map with a Conditional Kernel Given a Function of It

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $(Z,d_{Z})$, $(Y_{1},d_{1})$ and $(Y_{2},d_{2})$ be metric spaces with Borel $\sigma$-algebras $\mathcal{B}(Z)$, $\mathcal{B}(Y_{1})$ and $\mathcal{B}(Y_{2})$, and let $q:Z\to Y_{1}$ and $s:Y_{1}\to Y_{2}$ be measurable with respect to these $\sigma$-algebras, so that $s\circ q$ is measurable by claim 4 of Borel Measurability and Bounded Integration on a Metric Space. Let $\pi$ be a Borel measure on $(Z,d_{Z})$ with $\pi(Z)=1$, and let $\nu_{1}=q_{\#}\pi$ and $\nu_{2}=(s\circ q)_{\#}\pi$ be the image measures of claim 1 of that lemma, which are probability measures on $(Y_{1},\mathcal{B}(Y_{1}))$ and $(Y_{2},\mathcal{B}(Y_{2}))$, that is, Borel measures of total mass $1$ on $(Y_{1},d_{1})$ and $(Y_{2},d_{2})$. For every $A\in\mathcal{B}(Y_{2})$ one has $(s\circ q)^{-1}(A)=q^{-1}(s^{-1}(A))$, so $\nu_{2}(A)=\nu_{1}(s^{-1}(A))$; that is, $\nu_{2}=s_{\#}\nu_{1}$.

Let $\kappa$ be a conditional kernel of $\pi$ given $q$, and let $\lambda$ be a conditional kernel of $\nu_{1}$ given $s$, that is, one in the sense of that definition with $(Y_{1},d_{1})$, $(Y_{2},d_{2})$, $s$ and $\nu_{1}$ in place of $(Z,d_{Z})$, $(Y,d_{Y})$, $q$ and $\pi$, whose image measure there is $s_{\#}\nu_{1}=\nu_{2}$. Write $\kappa_{y}=\kappa(y,\cdot)$ and $\lambda_{w}=\lambda(w,\cdot)$. For $w\in Y_{2}$, $\lambda_{w}$ is a probability measure on $(Y_{1},\mathcal{B}(Y_{1}))$ by Probability Kernels Between Measurable Spaces §kernel, hence a Borel measure on $(Y_{1},d_{1})$ of finite mass; for $B\in\mathcal{B}(Z)$ the function $y\mapsto\kappa(y,B)$ on $Y_{1}$ is measurable with respect to $\mathcal{B}(Y_{1})$ and takes values in $[0,1]$ by the same definition, so it is integrable with respect to $\lambda_{w}$ by claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space. For a measure $\lambda'$ and a function $h$ integrable with respect to it, $\int h(y)\,\lambda'(dy)$ denotes $\int h\,d\lambda'$.

1. (Composition)¶ The function $\kappa\circ\lambda:Y_{2}\times\mathcal{B}(Z)\to\mathbb{R}$,

$$(\kappa\circ\lambda)(w,B)=\int_{Y_{1}}\kappa(y,B)\,\lambda_{w}(dy),$$

is a probability kernel from $(Y_{2},\mathcal{B}(Y_{2}))$ to $(Z,\mathcal{B}(Z))$ and a conditional kernel of $\pi$ given $s\circ q$.