Filtered probability space ($(\Omega ,\mathcal{F},({\mathcal{F}}_t),\mathbb{P})$)

A filtered probability space is a probability space together with an increasing family of sigma-algebras ${\mathcal{F}}_t\subseteq \mathcal{F}$ describing the information available by time $t$.

The usual conditions are completeness and right continuity of a filtration: ${\mathcal{F}}_0$ contains all subsets of null events in the ambient probability space, and ${\mathcal{F}}_t={\bigcap }_{s>t}{\mathcal{F}}_s$. They allow the standard continuous-time martingale, stopping time and stochastic integration theorems to be used without repeated augmentation qualifications. An absolute continuity of measures change preserves old null sets but can introduce new ones; the usual completion under the new measure may therefore be understood when needed.