Definition:Kernel Transformation of Positive Measurable Function
Jump to navigation
Jump to search
Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $N: X \times \Sigma \to \overline \R_{\ge 0}$ be a kernel.
Let $f: X \to \overline \R$ be a positive measurable function.
The transformation of $f$ by $N$ is the mapping $N f: X \to \overline \R$ defined by:
- $\forall x \in X: N \map f x := \ds \int f \rd N_x$
where $N_x$ is the measure $E \mapsto \map N {x, E}$.
Also see
- Kernel Transformation of Positive Measurable Function is Positive Measurable Function
- Kernel Transformation of Measure
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 9$: Problem $11 \ \text{(ii)}$