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