Definition:Kernel (Measure Theory)

Definition
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

Also see

 * Definition:Kernel Transformation of Measure
 * Definition:Kernel Transformation of Positive Measurable Function