Conditional interior and conditional closure of random sets
Résumé
In this short note, we present two new concepts. On a complete probability space, we consider two σ-algebras H ⊆ F and a F-graph-measurable random set Γ ⊆ R d. We show the existence of a largest H-measurable open set contained in X, we call conditional interior and a smallest H-measurable closed set containing X, we call conditional closure. We then deduce that a conditional essential supremum of real-valued random variables is actually a pointwise supremum over a closed random set.
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