Definition:Kernel (Measure Theory)

This page is about Kernel in the context of Measure Theory. For other uses, see Kernel.

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

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Also see

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

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 9$: Problem $11$