EN FR
EN FR

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

A process $\left\{{X}_{t},t\in T\right\}$ 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 $O\in 𝒪$,

$\left\{\omega :{X}_{s}\left(\omega \right)\in F\mathrm{for}\mathrm{all}s\in O\cap S\right\}\setminus \left\{\omega :{X}_{s}\left(\omega \right)\in F\mathrm{for}\mathrm{all}s\in O\right\}\subset \Lambda .$

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 $X=\left\{{X}_{t};\phantom{\rule{4pt}{0ex}}t\in T\right\}$ 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 $\left(T,{d}_{T}\right)$ is totally bounded, which is the case in