Kernel Transformation of Measure is Measure

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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.


Then $\mu N: X \to \overline{\R}$, the kernel transformation of $\mu$, is a measure.


Proof


Sources