I'm reading the probabilistic book write by çinlar, but I don't understand the Kernel theory, in details:
$ (E,\mathcal{E}),(F,\mathcal{F})$ are two measurable space
$$K:E \times \mathcal{F} \rightarrow R\_{+}$$
with these proprieties: they are measurable on $E$ fixed $B\in\mathcal{F}$, and they are a measure on $F$ fixed $x\in E$. After this definitio the book present the following operation between Kernels and functions:
$$Kf(x)=\int\_{F}K(x,dy)f(y)$$
But what does $K(x,dy)$ mean? in the theory of measure i haven't see nothing simalar, how one can calculate this integrals with the strange $dy$ (that I think means lebesgue measure)?



Theory of integration of Kernel in çinlar probability and stochastic
I've resolved! since the kernel is a measure fixed $x$ I can integrate the function $f$ respect to this measure (notation $\nu(dx)$ means integrate respet to measure $\nu$ and variables $x$)!


