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
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