Let $(\Omega_1,\mathcal{A}_1)$ and $(\Omega_2,\mathcal{A}_2)$ be measurable spaces. A map \begin{align} K: \Omega_1 \times \mathcal{F}_2 \to [0,+\infty] \end{align} is called a transition kernel from $\Omega_1$ to $\Omega_2$ if
- $K(.,A_2)$ is $\mathcal{A}_1$-measurable for any $A_2\in\mathcal{A}_2$.
- $K(\omega_1,.)$ is a measure on $(\Omega_2,\mathcal{A}_2)$ for any $\omega_1\in\mathcal{A}_1$.
We usually name the kernel according to the type of measures it creates. For instance, if the measures generated by $K$ are probability measures, $K$ is called a Markov kernel.
Also, we know that for Polish spaces and, in particular, real-valued processes, the kernels define regular conditional probabilities.
MCMC by Casella and Robert has the following equation $$ P((X_1, X_2)\in A_1\times A_2) = \int_{A_1}K(y,A_2)K(x,dy) $$
Clearly, we may write $$ P(x,A) = \int_A K(x,dy) $$ as the kernel defines a measure for fixed $x$. However, I'm having a hard time doing the computations for the cases with more than one variable that leads to the Chapman-Kolmogorov equations. I'd like a little help with them.