Given $A \in \sigma(\mathcal{S})$, with $\mathcal{S}$ is the state space, the transition kernel is a function $K(\cdot , \cdot): \mathcal{S} \times \mathcal{B}(\mathcal{S}) \to [0,1]$
- $\forall x \in \mathcal{S}, K(x,\cdot)$ is a probability measure;
- $\forall A \in \mathcal{B}(\mathcal{S}), K(\cdot,A)$ is measurable.
If the process at the initial time (t = 0) starts from a fixed state $x_0$: $$Pr\{(X_1,X_2) \in A_1 \times A_2\}=\int_{A_1 } K(y,A_2)K(x_0,dy)$$ is it possible to give an intuitive idea of the concepts: "probability measure" and "measurable" and then give an intuitive idea of the meaning of the following notation? $$\int_{A_1 } K(y,A_2)K(x_0,dy)$$ in particular of $$K(x_0,dy)$$ ?