In
mathematics and in particular
measure theory, a measurable function is a function between the underlying sets of two
measurable spaces that preserves the structure of the spaces: the
preimage of any
measurable
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 ...
set is measurable. This is in direct analogy to the definition that a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
function between
topological spaces
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
preserves
Fruit preserves are preparations of fruits whose main preserving agent is sugar and sometimes acid, often stored in glass jars and used as a condiment or spread.
There are many varieties of fruit preserves globally, distinguished by the meth ...
the topological structure: the preimage of any
open set
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 su ...
is open. In
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
, measurable functions are used in the definition of the
Lebesgue integral
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
. In
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 ...
, a measurable function on a
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
is known as a
random variable.
Formal definition
Let
and
be measurable spaces, meaning that
and
are sets equipped with respective
-algebras and
A function
is said to be measurable if for every
the pre-image of
under
is in
; that is, for all
That is,
where
is the
σ-algebra generated by f. If
is a measurable function, we will write
to emphasize the dependency on the
-algebras
and
Term usage variations
The choice of
-algebras in the definition above is sometimes implicit and left up to the context. For example, for
or other topological spaces, the
Borel algebra
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are nam ...
(generated by all the open sets) is a common choice. Some authors define measurable functions as exclusively real-valued ones with respect to the Borel algebra.
If the values of the function lie in an
infinite-dimensional vector space
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to di ...
, other non-equivalent definitions of measurability, such as
weak measurability and
Bochner measurability, exist.
Notable classes of measurable functions
* Random variables are by definition measurable functions defined on probability spaces.
* If
and
are
Borel spaces, a measurable function
is also called a Borel function. Continuous functions are Borel functions but not all Borel functions are continuous. However, a measurable function is nearly a continuous function; see
Luzin's theorem
In the mathematical field of real analysis, Lusin's theorem (or Luzin's theorem, named for Nikolai Luzin) or Lusin's criterion states that an almost-everywhere finite function is measurable if and only if it is a continuous function on nearly ...
. If a Borel function happens to be a section of a map
it is called a Borel section.
* A
Lebesgue measurable
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
function is a measurable function
where
is the
-algebra of Lebesgue measurable sets, and
is the
Borel algebra
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are nam ...
on the
complex number
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 fo ...
s
Lebesgue measurable functions are of interest in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
because they can be integrated. In the case
is Lebesgue measurable if and only if
is measurable for all
This is also equivalent to any of
being measurable for all
or the preimage of any open set being measurable. Continuous functions, monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functions of bounded variation are all Lebesgue measurable.
A function
is measurable if and only if the real and imaginary parts are measurable.
Properties of measurable functions
* The sum and product of two complex-valued measurable functions are measurable.
So is the quotient, so long as there is no division by zero.
* If
and
are measurable functions, then so is their composition
* If
and
are measurable functions, their composition
need not be
-measurable unless
Indeed, two Lebesgue-measurable functions may be constructed in such a way as to make their composition non-Lebesgue-measurable.
* The (pointwise)
supremum,
infimum
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 lo ...
,
limit superior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a ...
, and
limit inferior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For ...
of a sequence (viz., countably many) of real-valued measurable functions are all measurable as well.
*The
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
limit of a sequence of measurable functions
is measurable, where
is a metric space (endowed with the Borel algebra). This is not true in general if
is non-metrizable. Note that the corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence.
Non-measurable functions
Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to prove the existence of non-measurable functions. Such proofs rely on the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
in an essential way, in the sense that
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
without the axiom of choice does not prove the existence of such functions.
In any measure space ''
'' with a
non-measurable set
In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "volume". The mathematical existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory. In Zerm ...
one can construct a non-measurable
indicator function:
where
is equipped with the usual
Borel algebra
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are nam ...
. This is a non-measurable function since the preimage of the measurable set
is the non-measurable
As another example, any non-constant function
is non-measurable with respect to the trivial
-algebra
since the preimage of any point in the range is some proper, nonempty subset of
which is not an element of the trivial
See also
*
*
* - Vector spaces of measurable functions: the
spaces
*
*
*
Notes
External links
Measurable functionat
Encyclopedia of Mathematics
The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structu ...
Borel functionat
Encyclopedia of Mathematics
The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structu ...
{{DEFAULTSORT:Measurable Function
Measure theory
Types of functions