Markov kernel intution The usual definition of a Markov kernel (as for example the Wikipedia definition of a Markov kernel) introduces it as a map from the product space of a set (equipped with a sigma algebra) and another sigma algebra to the closed real unit interval. The common way this concept is thaught is by describing it as the continuos analog of a transition matrix.
The reason why it is not defined as a map from the product space of the two underlying base set of the sigma algebras is that the probability measure generated by the markov kernel needn't be defined for all singletons but it's enough to know their values for measurable sets.
But why is it not a map from the product space of the sigma algebras? Why do we need information about the exact element in one component of the kernel.

You should think about a Markov kernel as a non-deterministic generalized map from one space to another, that instead of assigning any element on the first space with an element on the second, it assigns any element in the first space with a probability measure on the second space, so it actually spreads each element of the space over the other space instead of sending it to a single element.
