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 ...
, the total variation identifies several slightly different concepts, related to the (
local or global) structure of the
codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
of a
function or a
measure. For a
real-valued continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
''f'', defined on an
interval 'a'', ''b''⊂ R, its total variation on the interval of definition is a measure of the one-dimensional
arclength of the curve with parametric equation ''x'' ↦ ''f''(''x''), for ''x'' ∈
'a'', ''b'' Functions whose total variation is finite are called
functions of bounded variation.
Historical note
The concept of total variation for functions of one real variable was first introduced by
Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''.
Biography
Jordan was born in Lyon and educated at ...
in the paper . He used the new concept in order to prove a convergence theorem for
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
of
discontinuous periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to des ...
s whose variation is
bounded. The extension of the concept to functions of more than one variable however is not simple for various reasons.
Definitions
Total variation for functions of one real variable
The total variation of a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
-valued (or more generally
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued)
function , defined on an
interval is the quantity
:
where the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
runs over 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 all
partitions
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of ...
of the given
interval.
Total variation for functions of ''n'' > 1 real variables
Let Ω be an
open subset
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of R
''n''. Given a function ''f'' belonging to ''L''
1(Ω), the total variation of ''f'' in Ω is defined as
:
where
*
is 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
continuously differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
vector functions of
compact support
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalle ...
contained in
,
*
is the
essential supremum
In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for ''all' ...
norm, and
*
is the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
operator.
This definition ''does not require'' that the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
of the given function be a
bounded set
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of m ...
.
Total variation in measure theory
Classical total variation definition
Following , consider a
signed measure
In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values.
Definition
There are two slightly different concepts of a signed measure, depending on whether or not ...
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 ...
: then it is possible to define two
set function
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R an ...
s
and
, respectively called upper variation and lower variation, as follows
:
:
clearly
:
The variation (also called absolute variation) of the signed measure
is the set function
:
and its total variation is defined as the value of this measure on the whole space of definition, i.e.
:
Modern definition of total variation norm
uses upper and lower variations to prove the
Hahn–Jordan decomposition: according to his version of this theorem, the upper and lower variation are respectively a
non-negative
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
and a
non-positive
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
measure. Using a more modern notation, define
:
:
Then
and
are two non-negative
measures such that
:
:
The last measure is sometimes called, by
abuse of notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors a ...
, total variation measure.
Total variation norm of complex measures
If the measure
is
complex-valued
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
i.e. is a
complex measure In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size (length, area, volume) is a complex number.
Definition
Formally ...
, its upper and lower variation cannot be defined and the Hahn–Jordan decomposition theorem can only be applied to its real and imaginary parts. However, it is possible to follow and define the total variation of the complex-valued measure
as follows
The variation of the complex-valued measure
is the
set function
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R an ...
:
where the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
is taken over all partitions
of a
measurable set into a countable number of disjoint measurable subsets.
This definition coincides with the above definition
for the case of real-valued signed measures.
Total variation norm of vector-valued measures
The variation so defined is a
positive measure
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 simil ...
(see ) and coincides with the one defined by when
is a
signed measure
In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values.
Definition
There are two slightly different concepts of a signed measure, depending on whether or not ...
: its total variation is defined as above. This definition works also if
is a
vector measure: the variation is then defined by the following formula
:
where the supremum is as above. This definition is slightly more general than the one given by since it requires only to consider ''finite partitions'' of the space
: this implies that it can be used also to define the total variation on
finite-additive measures.
Total variation of probability measures
The total variation of any
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more ge ...
is exactly one, therefore it is not interesting as a means of investigating the properties of such measures. However, when μ and ν are
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more ge ...
s, the
total variation distance of probability measures can be defined as
where the norm is the total variation norm of signed measures. Using the property that
, we eventually arrive at the equivalent definition
:
and its values are non-trivial. The factor
above is usually dropped (as is the convention in the article
total variation distance of probability measures). Informally, this is the largest possible difference between the probabilities that the two
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
s can assign to the same event. For a
categorical distribution
In probability theory and statistics, a categorical distribution (also called a generalized Bernoulli distribution, multinoulli distribution) is a discrete probability distribution that describes the possible results of a random variable that can ...
it is possible to write the total variation distance as follows
:
It may also be normalized to values in