This tag is for questions relating to "Filtrations". It has many application in abstract algebra, homological algebra, topology, measure theory and probability theory for nested sequences of $\sigma$-algebras.
A family $\{\mathcal{G}_t:t\geq0\}$ of sub-$\sigma$-algebras is called a filtration if $s<t$ implies $\mathcal{G}_s\subseteqq\mathcal{G}_t$.
The most natural filtration is the family of sub-$\sigma$-algebras $\mathcal{F}_t$’s which are minimally constructed so that $M(t)$ is just $\mathcal{F}_t$-measurable. Such a family of $\mathcal{F}_t$’s is viewed as the information generated by the process $M(t)$, and called a history of the process $M(t)$.
- A ring equipped with a filtration is called a filtered ring.
- The concept dual to a filtration is called a cofiltration.
Applications: Filtrations are widely used in abstract algebra, homological algebra (where they are related in an important way to spectral sequences), and in measure theory and probability theory for nested sequences of σ-algebras. In functional analysis and numerical analysis, other terminology is usually used, such as scale of spaces or nested spaces. In the theory of stochastic processes, a subdiscipline of probability theory, filtrations are used to model the information that is available at a given point and therefore play an important role in the formalization of random processes.
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