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
1105Keywords: Optional processes,
Separability,
Section theoremsNature: Exposition,
Original additions
Retrieve article from Numdam
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 timeNature: Original Retrieve article from Numdam
