Definition:Kernel (Measure Theory)

Definition
Let $\left({X, \Sigma}\right)$ be a measurable space.

Let $\overline{\R}_{\ge0}$ be the set of positive extended real numbers.

A kernel is a mapping $N: X \times \Sigma \to \overline{\R}_{\ge0}$ such that:


 * $(1):\quad \forall x \in X: N_x: \Sigma \to \overline{\R}_{\ge0}, E \mapsto N(x, E)$ is a measure
 * $(2):\quad \forall E \in \Sigma: N_E: X \to \overline{\R}_{\ge0}, x \mapsto N(x, E)$ is a positive $\Sigma$-measurable function

Also see

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