Definition:Kernel Transformation of Measure

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

Let $N: X \times \Sigma \to \overline{\R}_{\ge0}$ be a kernel.

The transformation of $\mu$ by $N$ is the mapping $\mu N: \Sigma \to \overline{\R}$ defined by:


 * $\displaystyle \forall E \in \Sigma: \mu N \left({E}\right) := \int N_E \left({x}\right) \, d\mu \left({x}\right)$

where $N_E \left({x}\right) = N \left({x, E}\right)$.

Also see

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