# Kernel Transformation of Positive Measurable Function is Positive Measurable Function

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## Theorem

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.

Then $N f: X \to \overline{\R}$, the transformation of $f$ by $N$, is also a positive measurable function.

## Proof

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## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $\S 9$: Problem $11 \ \text{(ii)}$