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2 matches found
X: 25, 521-531, LNM 511 (1976)
BENVENISTE, Albert
Séparabilité optionnelle, d'après Doob (General theory of processes)
A real valued function $f(t)$ admits a countable set $D$ as a separating set if the graph of $f$ is contained in the closure of its restriction to $D$. Doob's well known theorem asserts that every process $X$ has a modification all sample functions of which admit a common separating set $D$ (deterministic). It is shown that if $D$ is allowed to consist of (the values of) countably many stopping times, then every optional process is separable without modification. Applications are given
Comment: Doob's original paper appeared in Ann. Inst. Fourier, 25, 1975. See also 1105
Keywords: Optional processes, Separability, Section theorems
Nature: Exposition, Original additions
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XI: 05, 47-50, LNM 581 (1977)
DELLACHERIE, Claude
Deux remarques sur la séparabilité optionnelle (General theory of processes)
Optional separability was defined by Doob, Ann. Inst. Fourier, 25, 1975. See also Benveniste, 1025. The main remark in this paper is the following: given any optional set $H$ with countable dense sections, there exists a continuous change of time $(T_t)$ indexed by $[0,1[$ such that $H$ is the union of all graphs $T_t$ for $t$ dyadic. Thus Doob's theorem amounts to the fact that every optional process becomes separable in the ordinary sense once a suitable continuous change of time has been performed
Keywords: Optional processes, Separability, Changes of time
Nature: Original
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