Filtration (mathematics)

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In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with the index i running over some index set I that is a totally ordered set, subject to the condition that if i ≥ j in I then Si ⊆ Sj. The concept dual to a filtration is called a cofiltration.

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.

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[edit] Examples

[edit] Algebra

[edit] Groups

In algebra, filtrations are ordinarily indexed by N, the set of natural numbers. A filtration of a group G, is then a nested sequence Gn of normal subgroups of G (that is, for any n we have Gn+1 ⊆ Gn).

Given a group G and a filtration Gn, there is a natural way to define a topology on G, said to be associated to the filtration. A basis for this topology is the set of all translates of subgroups appearing in the filtration, that is, a subset of G is defined to be open if it is a union of sets of the form aGn, where a∈G and n is a natural number.

The topology associated to a filtration on a group G makes G into a topological group.

The topology associated to a filtration Gn on a group G is Hausdorff if and only if ∩Gn = {1}.

If two filtrations Gn and G′n are defined on a group G, then the identity map from G to G, where the first copy of G is given the Gn-topology and the second the G′n-topology, is continuous if and only if for any n there is an m such that Gm ⊆G′n, that is, if and only if the identity map is continuous at 1. In particular, the two filtrations define the same topology if and only if for any subgroup appearing in one there is a smaller or equal one appearing in the other.

[edit] Rings and modules

Given a ring R and an R-module M, a filtration of M is a decreasing sequence of submodules Mn. This is therefore a special case of the notion for groups, with the additional condition that the subgroups be submodules. The associated topology is defined as for groups.

An important special case is known as the I-adic topology (or J-adic, etc.) Let R be a commutative ring, and I an ideal of R.

Given an R-module M, the sequence InM of submodules of M forms a filtration of M. The I-adic topology on M is then the topology associated to this filtration. If M is just the ring R itself, we have defined the I-adic topology on R.

When R is given the I-adic topology, R becomes a topological ring. If an R-module M is then given the I-adic topology, it becomes a topological R-module, relative to the topology given on R.

[edit] Measure theory

In measure theory, in particular in martingale theory and the theory of stochastic processes, a filtration is an increasing sequence of σ-algebras on a measurable space. That is, given a measurable space (\Omega, \mathcal{F}), a filtration is a sequence of σ-algebras \{ \mathcal{F}_{t} \}_{t \geq 0} with \mathcal{F}_{t} \subseteq \mathcal{F} for each t and

t_{1} \leq t_{2} \implies \mathcal{F}_{t_{1}} \subseteq \mathcal{F}_{t_{2}}.

The exact range of the "times" t will usually depend on context: the set of values for t might be discrete or continuous, bounded or unbounded. For example,

t \in \{ 0, 1, \dots, N \}, \mathbb{N}_{0}, [0, T] \mbox{ or } [0, + \infty).

Similarly, a filtered probability space (also known as a stochastic basis) is a probability space with a filtration of its σ-algebra.

It is also useful (in the case of an unbounded index set) to define \mathcal{F}_{\infty} as the σ-algebra generated by the infinite union of the \mathcal{F}_{t}'s, which is contained in \mathcal{F}:

\mathcal{F}_{\infty} = \sigma\left(\bigcup_{t \geq 0} \mathcal{F}_{t}\right) \subseteq \mathcal{F}.

A σ-algebra defines the set of events that can be measured, which in a probability context is equivalent to events that can be discriminated, or "questions that can be answered at time t". Therefore a filtration is often used to represent the change in the set of events that can be measured, through gain or loss of information. A typical example is in mathematical finance, where a filtration represents the information available at each time t, and is more and more precise (the set of measurable events is staying the same or increasing) as information from the present becomes available.

[edit] References

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