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V: 29, 290-310, LNM 191 (1971)
WALSH, John B.
Some topologies connected with Lebesgue measure (Markov processes, General theory of processes, Potential theory)
It is a recurrent theme in the theory of stochastic processes that time sets of measure $0$ should be ignored. Thus topologies on the line which ignore sets of measure $0$ are useful. The main topic here is the so-called essential topology, used in the paper of Chung and Walsh 522 in the same volume
Comment: See Doob Bull. Amer. Math. Soc., 72, 1966. An important application in given by Walsh 623 in the next volume. See the paper 1025 of Benveniste. For the use of a different topology see Ito J. Math. Soc. Japan, 20, 1968
Keywords: Essential topology
Nature: Original
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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 topology
Nature: Original
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VI: 23, 243-252, LNM 258 (1972)
MEYER, Paul-André
Quelques autres applications de la méthode de Walsh (La perfection en probabilités'') (Markov processes)
This is but an exercise on using the method of the preceding paper 622 to reduce the exceptional sets in other situations: additive functionals, cooptional times and processes, etc
Comment: A correction to this paper is mentioned on the errata list of vol. VII
Keywords: Additive functionals, Return times, Essential topology
Nature: Original
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