In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a filtration
is an
indexed family
In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a ''family of real numbers, indexed by the set of integers'' is a collection of real numbers, wher ...
of
subobjects of a given
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...
, with the index
running over some
totally ordered
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive ...
index set
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists ...
, subject to the condition that
::if
in
, then
.
If the index
is the time parameter of some
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
, then the filtration can be interpreted as representing all historical but not future information available about the stochastic process, with the algebraic structure
gaining in complexity with time. Hence, a process that is
adapted to a filtration
is also called non-anticipating, because it cannot "see into the future".
Sometimes, as in a
filtered algebra, there is instead the requirement that the
be
subalgebras with respect to some operations (say,
vector addition), but not with respect to other operations (say, multiplication) that satisfy only
, where the index set is the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s; this is by analogy with a
graded algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
.
Sometimes, filtrations are supposed to satisfy the additional requirement that the
union of the
be the whole
, or (in more general cases, when the notion of union does not make sense) that the canonical
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
from the
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...
of the
to
is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. Whether this requirement is assumed or not usually depends on the author of the text and is often explicitly stated. This article does ''not'' impose this requirement.
There is also the notion of a descending filtration, which is required to satisfy
in lieu of
(and, occasionally,
instead of
). Again, it depends on the context how exactly the word "filtration" is to be understood. Descending filtrations are not to be confused with the
dual notion of cofiltrations (which consist of
quotient object In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory ...
s rather than
subobject In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory ...
s).
Filtrations are widely used in
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
,
homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topolo ...
(where they are related in an important way to
spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they h ...
s), and in
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
and
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
for nested sequences of
σ-algebras. In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
and
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
, other terminology is usually used, such as
scale of spaces or
nested spaces.
Examples
Algebra
Algebras
See:
Filtered algebra
Groups
In algebra, filtrations are ordinarily indexed by
, the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of natural numbers. A ''filtration'' of a group
, is then a nested sequence
of
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
s of
(that is, for any
we have
). Note that this use of the word "filtration" corresponds to our "descending filtration".
Given a group
and a filtration
, there is a natural way to define a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
on
, said to be ''associated'' to the filtration. A basis for this topology is the set of all
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s of subgroups appearing in the filtration, that is, a subset of
is defined to be open if it is a union of sets of the form
, where
and
is a natural number.
The topology associated to a filtration on a group
makes
into a
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
.
The topology associated to a filtration
on a group
is
Hausdorff if and only if
.
If two filtrations
and
are defined on a group
, then the identity map from
to
, where the first copy of
is given the
-topology and the second the
-topology, is continuous if and only if for any
there is an
such that
, 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.
Rings and modules: descending filtrations
Given a ring
and an
-module
, a ''descending filtration'' of
is a decreasing sequence of
submodule
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
s
. 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
-adic topology (or
-adic, etc.): Let
be a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, and
an ideal of
. Given an
-module
, the sequence
of submodules of
forms a filtration of
. The ''
-adic topology'' on
is then the topology associated to this filtration. If
is just the ring
itself, we have defined the ''
-adic topology'' on
.
When
is given the
-adic topology,
becomes a
topological ring In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps:
R \times R \to R
where R \times R carries the product topology. That means R is an additive ...
. If an
-module
is then given the
-adic topology, it becomes a
topological -module, relative to the topology given on
.
Rings and modules: ascending filtrations
Given a ring
and an
-module
, an ''ascending filtration'' of
is an increasing sequence of submodules
. In particular, if
is a field, then an ascending filtration of the
-vector space
is an increasing sequence of
vector subspaces of
.
Flags are one important class of such filtrations.
Sets
A maximal filtration of a set is equivalent to an ordering (a
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
) of the set. For instance, the filtration
corresponds to the ordering
. From the point of view of the
field with one element
In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French–English pun, Fun. The name ...
, an ordering on a set corresponds to a maximal
flag
A flag is a piece of fabric (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design empl ...
(a filtration on a vector space), considering a set to be a vector space over the field with one element.
Measure theory
In
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
, in particular in
martingale theory
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all p ...
and the theory of
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
es, a filtration is an increasing
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of
-algebras on a
measurable space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured.
Definition
Consider a set X and a σ-algebra \mathcal A on X. Then the ...
. That is, given a measurable space
, a filtration is a sequence of
-algebras
with
where each
is a non-negative
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
and
:
The exact range of the "times" ''
'' will usually depend on context: the set of values for
might be
discrete or continuous,
bounded or unbounded. For example,
:
Similarly,_a_filtered_probability_space_(also_known_as_a_stochastic_basis)_
\left(\Omega,_\mathcal,_\left\_,_\mathbb\right),_is_a_probability_space_equipped_with_the_filtration_
\left\__of_its_
\sigma-algebra_
\mathcal._A_filtered_probability_space_is_said_to_satisfy_the_''usual_conditions''_if_it_is_complete_measure.html" "title="probability_space.html" ;"title=", + \infty).
Similarly, a filtered probability space (also known as a stochastic basis)
\left(\Omega, \mathcal, \left\_, \mathbb\right), is a probability space">, + \infty).
Similarly, a filtered probability space (also known as a stochastic basis)
\left(\Omega, \mathcal, \left\_, \mathbb\right), is a probability space equipped with the filtration
\left\_ of its
\sigma-algebra
\mathcal. A filtered probability space is said to satisfy the ''usual conditions'' if it is complete measure">complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
-null sets) and right-continuous (i.e.
).
context is equivalent to events that can be discriminated, or "questions that can be answered at time
". Therefore, a filtration is often used to represent the change in the set of events that can be measured, through gain or loss of
. A typical example is in
, where a filtration represents the information available up to and including each time
, and is more and more precise (the set of measurable events is staying the same or increasing) as more information from the evolution of the stock price becomes available.
be a filtered probability space. A random variable