Lebesgue Measurable Function
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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 ...
and in particular
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 simil ...
, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
of any
measurable In mathematics, the concept of a measure is a generalization and formalization of Geometry#Length, area, and volume, geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly ...
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 points ...
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 method ...
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 suf ...
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 converg ...
, 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 o ...
, 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 A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
.


Formal definition

Let (X,\Sigma) and (Y,\Tau) be measurable spaces, meaning that X and Y are sets equipped with respective \sigma-algebras \Sigma and \Tau. A function f:X\to Y is said to be measurable if for every E\in \Tau the pre-image of E under f is in \Sigma; that is, for all E \in \Tau f^(E) := \ \in \Sigma. That is, \sigma (f)\subseteq\Sigma, where \sigma (f) is the σ-algebra generated by f. If f:X\to Y is a measurable function, we will write f \colon (X, \Sigma) \rightarrow (Y, \Tau). to emphasize the dependency on the \sigma-algebras \Sigma and \Tau.


Term usage variations

The choice of \sigma-algebras in the definition above is sometimes implicit and left up to the context. For example, for \R, \Complex, 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 named ...
(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 (X, \Sigma) and (Y, T) are Borel spaces, a measurable function f:(X, \Sigma) \to (Y, T) 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 Y\xrightarrowX, 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 f : (\R, \mathcal) \to (\Complex, \mathcal_\Complex), where \mathcal is the \sigma-algebra of Lebesgue measurable sets, and \mathcal_\Complex 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 named ...
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 form ...
s \Complex. 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 f : X \to \R, f is Lebesgue measurable if and only if \ = \ is measurable for all \alpha\in\R. This is also equivalent to any of \,\,\ being measurable for all \alpha, 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 f:X\to\Complex 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 f : (X,\Sigma_1) \to (Y,\Sigma_2) and g:(Y,\Sigma_2) \to (Z,\Sigma_3) are measurable functions, then so is their composition g\circ f:(X,\Sigma_1) \to (Z,\Sigma_3). * If f : (X,\Sigma_1) \to (Y,\Sigma_2) and g:(Y,\Sigma_3) \to (Z,\Sigma_4) are measurable functions, their composition g\circ f: X\to Z need not be (\Sigma_1,\Sigma_4)-measurable unless \Sigma_3 \subseteq \Sigma_2. Indeed, two Lebesgue-measurable functions may be constructed in such a way as to make their composition non-Lebesgue-measurable. * The (pointwise)
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 l ...
,
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 low ...
,
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 f_n: X \to Y is measurable, where Y is a metric space (endowed with the Borel algebra). This is not true in general if Y 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 collectio ...
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 as ...
without the axiom of choice does not prove the existence of such functions. In any measure space ''(X, \Sigma)'' 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 ...
A \subset X, A \notin \Sigma, one can construct a non-measurable
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
: \mathbf_A:(X,\Sigma) \to \R, \quad \mathbf_A(x) = \begin 1 & \text x \in A \\ 0 & \text, \end where \R 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 named ...
. This is a non-measurable function since the preimage of the measurable set \ is the non-measurable A.   As another example, any non-constant function f : X \to \R is non-measurable with respect to the trivial \sigma-algebra \Sigma = \, since the preimage of any point in the range is some proper, nonempty subset of X, which is not an element of the trivial \Sigma.


See also

* * * - Vector spaces of measurable functions: the L^p spaces * * *


Notes


External links


Measurable function
at
Encyclopedia of Mathematics The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics. Overview The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduat ...

Borel function
at
Encyclopedia of Mathematics The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics. Overview The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduat ...
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