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 $(T,𝒪)$ be a topological space. We assume that this space is second-countable, ie there exists a countable subset $\widetilde{𝒪}\subseteq 𝒪$ such that any open set of $𝒪$ can be expressed as a union of elements of $\widetilde{𝒪}$.

A process $\{X_t,t\in T\}$ is separable if there exists an at most countable set $S\subset T$ and a null set $\Lambda$ such that for all closed sets $F\subset 𝐑$ and all open set $O\in 𝒪$,

This definition is different of the one found in [57] , where the space is “linear", in that this author considers the previous equation only when $O$ 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 $T$-indexed stochastic process $X=\{X_t;t\in T\}$ has a separable modification.

If $T$ is an indexing collection in the sense of [62] , the topology induced by the distance $d_T$ has to be second-countable. This happens for instance when $(T,d_T)$ is totally bounded, which is the case in