Integral with respect to Kernel Transformation of Measure

Theorem
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.

Then:


 * $\displaystyle \int f \map \rd {\mu N} = \int N f \rd \mu$

where:


 * The integral sign denotes integration with respect to a measure
 * $\mu N$ is the transformation of $\mu$ by $N$
 * $N f$ is the transformation of $f$ by $N$

Writing $\map \mu f$ in place of $\displaystyle \int f \rd \mu$, the theorem statement can be conveniently expressed as:


 * $\map {\mu N} f = \map \mu {N f}$