XII: 35, 482-488, LNM 649 (1978)
YOR, Marc;
MEYER, Paul-André
Sur l'extension d'un théorème de Doob à un noyau $\sigma$-fini, d'après G. Mokobodzki (
Measure theory)
Given a kernel $K(x,dy)$ consisting of probability measures, all of them absolutely continuous with respect to a measure $\mu$, Doob proved long ago using martingale theory that $K(x,dy)=k(x,y)\,\mu(dy)$ with a jointly measurable density $k(x,y)$. What happens if the measures $K(x,dy)$ are $\sigma$-finite? The answer is that Doob's result remains valid if $K$, considered as a mapping $x\mapsto K(x,\,.\,)$ taking values in the set of all $\sigma$-finite measures absolutely continuous w.r.t. $\mu$ (i.e., of classes of a.s. finite measurable functions), is Borel with respect to the topology of convergence in probability
Comment: The subject is discussed further in
1527. Note a mistake near the bottom of page 486: the $\sigma$-field on $E$ should be associated with the
weak topology of $L[\infty$, or with the topology of $L^0$
Keywords: Kernels,
Radon-Nikodym theoremNature: Original
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