A \(D_E[0,1]\) representation of random upper semicontinuous functions. (English) Zbl 1005.28003
Summary: A representation of random upper semicontinuous functions in terms of \(D_E[0,1]\)-valued random elements is stated. This fact allows us to consider for the first time a complete and separable metric, the Skorohod one, on a wide class of upper semicontinuous functions. Finally, different relevant concepts of measurability for random upper semicontinuous functions are studied and the relationships between them are analyzed.
MSC:
| 28A20 | Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence |
| 54E50 | Complete metric spaces |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 60B99 | Probability theory on algebraic and topological structures |
| 60F05 | Central limit and other weak theorems |
| 54C35 | Function spaces in general topology |
Keywords:
cadlag function; measurability; random upper semicontinuous function; Skorohod metric; uniform metricReferences:
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