Integration Against a Probability Kernel: Measurable Sections, the Composite Measure on the Product and the Iterated Integral

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $(Y,\mathcal{Y})$ and $(Z,\mathcal{Z})$ be measurable spaces, let $\kappa$ be a probability kernel from $(Y,\mathcal{Y})$ to $(Z,\mathcal{Z})$, with $\kappa_{y}=\kappa(y,\cdot)$, and let $\mu$ be a probability measure on $(Y,\mathcal{Y})$. Let $\mathcal{Y}\otimes\mathcal{Z}$ be the product $\sigma$-algebra, and for $E\subseteq Y\times Z$ and $y\in Y$ let $E_{y}=\{z\in Z:(y,z)\in E\}$ be the section of $E$ at $y$; for $E\in\mathcal{Y}\otimes\mathcal{Z}$ it belongs to $\mathcal{Z}$ by claim 1 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable, and for $f:Y\times Z\to\mathbb{R}$ measurable with respect to $\mathcal{Y}\otimes\mathcal{Z}$ and $y\in Y$ the function $f(y,\cdot)$ is measurable with respect to $\mathcal{Z}$ by claim 3 of that lemma. For a measure $\lambda$ on a measurable space and an integrable or nonnegative measurable function $h$ on it we also write $\int h(t)\,\lambda(dt)$ for $\int h\,d\lambda$, and $\mathbf{1}_{A}$ is the indicator function of a set $A$.

1. (Sections)¶ For every $E\in\mathcal{Y}\otimes\mathcal{Z}$ the function $y\mapsto\kappa(y,E_{y})$ on $Y$ takes values in $[0,1]$, is measurable with respect to $\mathcal{Y}$, and is integrable with respect to $\mu$.

2. (Composite measure)¶ The function $\mu\otimes\kappa:\mathcal{Y}\otimes\mathcal{Z}\to\mathbb{R}$,

$$(\mu\otimes\kappa)(E)=\int_{Y}\kappa(y,E_{y})\,\mu(dy),$$

is a probability measure on $(Y\times Z,\mathcal{Y}\otimes\mathcal{Z})$, and for all $A\in\mathcal{Y}$ and $B\in\mathcal{Z}$

$$(\mu\otimes\kappa)(A\times B)=\int_{Y}\mathbf{1}_{A}(y)\,\kappa(y,B)\,\mu(dy).$$

3. (Rectangles determine it)¶ If $\pi$ is a measure on $(Y\times Z,\mathcal{Y}\otimes\mathcal{Z})$ with $\pi(A\times B)=\int_{Y}\mathbf{1}_{A}(y)\,\kappa(y,B)\,\mu(dy)$ for all $A\in\mathcal{Y}$ and $B\in\mathcal{Z}$, then $\pi=\mu\otimes\kappa$.

4. (Bounded functions)¶ Let $f:Y\times Z\to\mathbb{R}$ be measurable with respect to $\mathcal{Y}\otimes\mathcal{Z}$ and bounded. Then for every $y\in Y$ the function $f(y,\cdot)$ is integrable with respect to $\kappa_{y}$, the function $g:Y\to\mathbb{R}$, $g(y)=\int_{Z}f(y,z)\,\kappa_{y}(dz)$, is measurable with respect to $\mathcal{Y}$ and bounded, $f$ is integrable with respect to $\mu\otimes\kappa$, $g$ is integrable with respect to $\mu$, and

$$\int_{Y\times Z}f\,d(\mu\otimes\kappa)=\int_{Y}g\,d\mu .$$

5. (Nonnegative functions)¶ Let $f:Y\times Z\to\mathbb{R}$ be measurable with respect to $\mathcal{Y}\otimes\mathcal{Z}$, with $f\ge0$. Then the set $N$ of the $y\in Y$ for which $f(y,\cdot)$ is not integrable with respect to $\kappa_{y}$ belongs to $\mathcal{Y}$, the function $g:Y\to\mathbb{R}$ equal to $\int_{Z}f(y,z)\,\kappa_{y}(dz)$ for $y\notin N$ and to $0$ for $y\in N$ is measurable with respect to $\mathcal{Y}$, and $f$ is integrable with respect to $\mu\otimes\kappa$ if and only if $\mu(N)=0$ and $g$ is integrable with respect to $\mu$; in that case

$$\int_{Y\times Z}f\,d(\mu\otimes\kappa)=\int_{Y}g\,d\mu .$$

6. (Constant kernels)¶ If there is a probability measure $\lambda$ on $(Z,\mathcal{Z})$ with $\kappa_{y}=\lambda$ for every $y\in Y$, then $\mu\otimes\kappa$ is the product measure $\mu\otimes\lambda$.