Kernels

Let \(E\) be a set and  \(\mathscr{E}\)  a \(\sigma\)-algebra of subsets of  \(E\). Assume that the
 \(\sigma\)-algebra  \(\mathscr{E}\) is  countably generated, i.e. generated by a countable collection
of subsets of \(E\).

The measurable space  \((E, \mathscr{E})\) is called the  state space and the
points of \(E\) are called  states.

The symbol  \(\mathscr{E}\) will also be used to denote the
collection of extended real valued measurable functions on  \((E, \mathscr{E})\).

The symbols \(x, y,\ldots\) denote states,  \(A,  B, \ldots\)  denote elements of the \(\sigma\)-algebra  \(\mathscr{E}\),
and \(f,g,\ldots\)  denote extended real valued measurable functions on  \((E,  \mathscr{E})\).