  In this note, we develop some of the basic theory of s-finite (measures and)
kernels, a little-studied class of kernels which Staton has recently
convincingly argued is precisely the semantic counterpart of (first-order)
probabilistic programs. We discuss their Carath\'eodory extension and extend
Staton's analysis of their product measures. We give various characterisations
of such kernels and discuss their relationship to the more commonly studied
classes of $\sigma$-finite, subprobability and probability kernels. We use
these characterisations to establish suitable Radon-Nikod\'ym, Lebesgue
decomoposition and disintegration theorems for s-finite kernels. We discuss
s-finite analogues of the classical randomisation lemma for probability
kernels. Throughout, we give some examples to explain the connection with
(first-order) probabilistic programming. Finally, we briefly explore how some
of these results extend to quasi-Borel spaces, hence how they apply to
higher-order probabilistic programming.
