XV: 27, 371-387, LNM 850 (1981)
DELLACHERIE, Claude
Sur les noyaux $\sigma$-finis (Measure theory)
This paper is an improvement of 1235. Assume $(X,{\cal X})$ and $(Y,{\cal Y})$ are measurable spaces and $m(x,A)$ is a kernel, i.e., is measurable in $x\in X$ for $A\in{\cal Y}$, and is a $\sigma$-finite measure in $A$ for $x\in X$. Then the problem is to represent the measures $m(x,dy)$ as $g(x,y)\,N(x,dy)$ where $g$ is a jointly measurable function and $N$ is a Markov kernel---possibly enlarging the $\sigma$-field ${\cal X}$ to include analytic sets. The crucial hypothesis (called measurability of $m$) is the following: for every auxiliary space $(Z, {\cal Z})$, the mapping $(x,z)\mapsto m_x\otimes \epsilon_z$ is again a kernel (in fact, the auxiliary space $R$ is all one needs). The case of ``basic'' kernels, considered in 1235, is thoroughly discussed
Keywords: Kernels, Radon-Nikodym theorem
Nature: Original
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