Definition:Kernel Transformation of Measure

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

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