\begin{definition}[Definition:Kernel (Measure Theory)]
Let $\struct {X, \Sigma}$ be a measurable space.
Let $\overline \R_{\ge 0}$ be the set of positive extended real numbers.
A '''kernel''' is a mapping $N: X \times \Sigma \to \overline{\R}_{\ge0}$ such that:
:$(1): \quad \forall x \in X: N_x: \Sigma \to \overline \R_{\ge 0}, E \mapsto \map N {x, E}$ is a measure
:$(2): \quad \forall E \in \Sigma: N_E: X \to \overline \R_{\ge 0}, x \mapsto \map N {x, E}$ is a positive $\Sigma$-measurable function
{{expand|I found a more general approach on [http://www.physicsforums.com/showthread.php?t{{=}}352389 this website]}}
\end{definition}