\begin{definition}[Definition:Kernel Transformation of Measure]
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 $\map {N_E} x = \map N {x, E}$.
\end{definition}