III: 05, 97-114, LNM 88 (1969)
DELLACHERIE, Claude
Ensembles aléatoires I (
Descriptive set theory,
Markov processes,
General theory of processes)
A deep theorem of Lusin asserts that a Borel set with countable sections is a countable union of Borel graphs. It is applied here in the general theory of processes to show that an optional set with countable sections is a countable union of graphs of stopping times, and in the theory of Markov processes, that a Borel set which is a.s. hit by the process at countably many times must be semi-polar
Comment: See Dellacherie,
Capacités et Processus Stochastiques, Springer 1972
Keywords: Sets with countable sectionsNature: Original
Retrieve article from Numdam
XII: 37, 491-508, LNM 649 (1978)
MOKOBODZKI, Gabriel
Ensembles à coupes dénombrables et capacités dominées par une mesure (
Measure theory,
General theory of processes)
Let $X$ be a compact metric space $\mu$ be a bounded measure. Let $F$ be a given Borel set in $X\times
R_+$. For $A\subset X$ define $C(A)$ as the outer measure of the projection on $X$ of $F\cap(A\times
R_+)$. Then it is proved that, if there is some measure $\lambda$ such that $\lambda$-null sets are $C$-null (the relation goes the reverse way from the preceding paper
1236!) then $F$ has ($\mu$-a.s.) countable sections, and if the property is strengthened to an $\epsilon-\delta$ ``absolute continuity'' relation, then $F$ has ($\mu$-a.s.) finite sections
Comment: This was a long-standing conjecture of Dellacherie (
707), suggested by the theory of semi-polar sets. For further development see
1602Keywords: Sets with countable sectionsNature: Original Retrieve article from Numdam
XII: 38, 509-511, LNM 649 (1978)
DELLACHERIE, Claude
Appendice à l'exposé de Mokobodzki (
Measure theory,
General theory of processes)
Some comments on
1237: a historical remark, a relation with a result of Talagrand, the inclusion of a converse (due to Horowitz) to the case of finite sections, and the solution to the conjecture from
707Keywords: Sets with countable sections,
Semi-polar setsNature: Original Retrieve article from Numdam
