# Definition:Kernel Transformation of Positive Measurable Function

## 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 $E \mapsto \map N {x, E}$.