Let $(I, \leq)$ be a totally ordered set. A family of sigma-algebras $\set{F_i}_{i \in I}$ is a filtration if $F_{j} \subset F_{k}$ for all $j \leq k$.
We call a filtration whose index set is the
natural numbers with their usual order a
discrete-time filtration.
We call a filtration whose index set is the
real numbers with their usual order a
continuous-time filtration.
The index set may also be finite, for example,
$\set{F_i}_{i = 1}^{n}$, in which case
\[
F_1 \subset F_2 \subset \cdots \subset F_n.
\]
It is extremely common to see filtrations written with the caligraphic $\mathcal{F} $. As in, let $\set{\mathcal{F} _{i}}_{i=1}^{n}$ be a filtration. This is in accordance with using caligraphic letters for sets of sets, and in accordance with the term sigma field for sigma algebra.