For an ordered set $T$ a filtration of $\sigma$ algebras $(\mathcal{F}_t)_{t\in T}$ is a collection of $\sigma$ algebras on an underlying set $\Omega$ satisfying $\mathcal{F}_s\subseteq\mathcal{F}_t$ for all $s<t$ in $T$ Here, $t$ is understood as the time variable, taking values in the index set $T$ and $\mathcal{F}_t$ represents the collection of all events observable up until time $t$ The index set is usually a subset of the real numbers, with common examples being $T=\mathbb{Z}_+$ for discrete-time and $T=\mathbb{R}_+$ for continuous-time scenarios. The collection $(\mathcal{F}_t)_{t\in T}$ is a filtration on a measurable space $(\Omega,\mathcal{F})$ if $\mathcal{F}_t\subseteq\mathcal{F}$ for every $t$ If, furthermore, there is a probability measure defined on the underlying measurable space then this gives a filtered probability space. The alternative notation $(\mathcal{F}_t,t\in T)$ is often used for the filtration or, when the index set $T$ is clear from the context, simply $(\mathcal{F}_t)$ or ${\bf F}$

Filtrations are widely used for studying stochastic processes, where a process $X_t$ with time ranging over the set $T$ is said to be adapted to the filtration if $X_t$ is an $\mathcal{F}_t$ measurable random variable for each time $t$

Conversely, any stochastic process $(X_t)_{t\in T}$ generates a filtration. Let $\mathcal{F}_t$ be the smallest $\sigma$ algebra with respect to which $X_s$ is measurable for all $s\le t$ \begin{equation*} \mathcal{F}_t=\sigma\left(X_s:s\le t\right). \end{equation*}This defines the smallest filtration to which $X$ is adapted, known as the natural filtration of $X$

Given a filtration, there are various limiting $\sigma$ algebras which can be defined. The values at plus and minus infinity are \begin{equation*} \mathcal{F}_\infty = \sigma\left(\bigcup_t\mathcal{F}_t\right),\ \mathcal{F}_{-\infty} = \bigcap_t\mathcal{F}_t, \end{equation*}which satisfy $\mathcal{F}_{-\infty}\subseteq\mathcal{F}_t\subseteq\mathcal{F}_\infty$ In continuous-time, when the index set is an interval of the real numbers, the left and right limits can be defined at any time. They are, \begin{equation*} \mathcal{F}_{t+}=\bigcap_{s>t}\mathcal{F}_s,\ \mathcal{F}_{t-}=\sigma\left(\bigcup_{s<t}\mathcal{F}_s\right), \end{equation*}except if $t$ is the maximum of $T$ it is often convenient to set $\mathcal{F}_{t+}=\mathcal{F}_t$ or, if $t$ is the minimum, $\mathcal{F}_{t-}=\mathcal{F}_t$ It is easily verified that $\mathcal{F}_s\subseteq\mathcal{F}_{s+}\subseteq\mathcal{F}_{t-}\subseteq\mathcal{F}_t$ for all times $s<t$ Furthermore, $(\mathcal{F}_{t+})$ and $(\mathcal{F}_{t-})$ are themselves filtrations.

A filtration is said to be right-continuous if $\mathcal{F}_t=\mathcal{F}_{t+}$ for every $t$ so, in particular, $(\mathcal{F}_{t+})$ is always the smallest right-continuous filtration larger than $(\mathcal{F}_t)$