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XII: 39, 512-514, LNM 649 (1978)

**DELLACHERIE, Claude**

Sur l'existence de certains ess.inf et ess.sup de familles de processus mesurables (General theory of processes)

The word ``essential'' in the title refers to inequalities between processes up to evanescent sets. Since in the case of a probability space consisting of one point, this means inequalities everywhere, it is clear that additional assumptions are necessary. Such essential bounds are shown to exist whenever the sample functions are upper semicontinuous in the right topology, or the left topology (and of course also if they are lower semicontinuous). This covers in particular the case of strong supermartingales and Snell's envelopes

Keywords: Essential suprema, Evanescent sets

Nature: Original

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XII: 43, 564-566, LNM 649 (1978)

**DELLACHERIE, Claude**; **MOKOBODZKI, Gabriel**

Deux propriétés des ensembles minces (abstraits) (Descriptive set theory)

Given a class ${\cal S}$ of Borel sets understood as ``small'' sets, the class ${\cal L}$ consisting of their conplements understood as ``large'' sets, a set $A$ is said to be ${\cal S}$-thin if does not contain uncountably many disjoint ``large'' sets. For instance, if ${\cal S}$ is the class of polar sets, then thin sets are the same as semi-polar sets. Two general theorems are proved here on thin sets

Keywords: Thin sets, Semi-polar sets, Essential suprema

Nature: Original

Retrieve article from Numdam

Sur l'existence de certains ess.inf et ess.sup de familles de processus mesurables (General theory of processes)

The word ``essential'' in the title refers to inequalities between processes up to evanescent sets. Since in the case of a probability space consisting of one point, this means inequalities everywhere, it is clear that additional assumptions are necessary. Such essential bounds are shown to exist whenever the sample functions are upper semicontinuous in the right topology, or the left topology (and of course also if they are lower semicontinuous). This covers in particular the case of strong supermartingales and Snell's envelopes

Keywords: Essential suprema, Evanescent sets

Nature: Original

Retrieve article from Numdam

XII: 43, 564-566, LNM 649 (1978)

Deux propriétés des ensembles minces (abstraits) (Descriptive set theory)

Given a class ${\cal S}$ of Borel sets understood as ``small'' sets, the class ${\cal L}$ consisting of their conplements understood as ``large'' sets, a set $A$ is said to be ${\cal S}$-thin if does not contain uncountably many disjoint ``large'' sets. For instance, if ${\cal S}$ is the class of polar sets, then thin sets are the same as semi-polar sets. Two general theorems are proved here on thin sets

Keywords: Thin sets, Semi-polar sets, Essential suprema

Nature: Original

Retrieve article from Numdam