Section: New Results
Separability of Set-Indexed Processes
Participant : Alexandre Richard.
A classical result states that any (multiparameter) stochastic process has a separable modification, thus ensuring the measurability property of the sample paths. We extend this result to set-indexed processes.
Let be a topological space. We assume that this space is second-countable, ie there exists a countable subset such that any open set of can be expressed as a union of elements of .
A process is separable if there exists an at most countable set and a null set such that for all closed sets and all open set ,
This definition is different of the one found in  , where the space is “linear", in that this author considers the previous equation only when is an interval. It happens that this notion needs not be defined in a general topological space. However when restricted to a vector space, our definition implies the previous one.
Theorem 0.2 (Doob's separability theorem) Any -indexed stochastic process has a separable modification.
If is an indexing collection in the sense of  , the topology induced by the distance has to be second-countable. This happens for instance when is totally bounded, which is the case in