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

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 $F\subset \mathbf{R}$ and all open set $O\in \mathcal{O}$,

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 $X=\{{X}_{t};\phantom{\rule{4pt}{0ex}}t\in T\}$ 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