VI: 22, 233-242, LNM 258 (1972)
WALSH, John B.
The perfection of multiplicative functionals (
Markov processes)
In the definition of multiplicative functionals the problem arose from the beginning whether the exceptional null set in the relation $M_{s+t}=M_s\,M_t\circ\theta_s$ was allowed to depend on $s$ or not---in the latter case the functional is said to be perfect. C.~Doléans showed by a detailed analysis (see
203) that every functional has a perfect modification, see also Dellacherie
304. Here a perfect version is constructed directly as $\lim_{s\rightarrow 0} M_{t-s}\circ\theta_s$, the limit being taken in the essential topology of the line, which ignores sets of zero Lebesgue measure
Keywords: Multiplicative functionals,
Perfection,
Essential topologyNature: Original
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