Definition:Kernel Transformation of Positive Measurable Function

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

Let $N: X \times \Sigma \to \overline{\R}_{\ge0}$ 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:


 * $\forall x \in X: N f \left({x}\right) := \displaystyle \int f \, \mathrm dN_x$

where $N_x$ is the measure $E \mapsto N(x, E)$.

Also see

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