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| In [[mathematics]], particularly in [[measure theory]], '''measurable functions''' are [[morphism|structure-preserving functions]] between [[measurable space]]s; as such, they form a natural context for the [[integral|theory of integration]]. Specifically, a function between measurable spaces is said to be '''measurable''' if the [[preimage]] of each [[measurable set]] is [[measurable]], analogous to the situation of [[continuity (topology)|continuous]] functions between [[topological space]]s.
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| This definition can be deceptively simple, however, as special care must be taken regarding the [[Sigma-algebra|σ-algebras]] involved. In particular, when a function ''f'': '''R''' → '''R''' is said to be [[Lebesgue measurable]] what is actually meant is that <math>f : (\mathbf{R}, \mathcal{L}) \to (\mathbf{R}, \mathcal{B})</math> is a measurable function—that is, the domain and range represent different σ-algebras on the same underlying set (here <math>\mathcal{L}</math> is the [[sigma algebra]] of [[Lebesgue measurable]] sets, and <math>\mathcal{B}</math> is the [[Borel algebra]] on '''R'''). As a result, the composition of Lebesgue-measurable functions need not be Lebesgue-measurable.
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| By convention a [[topological space]] is assumed to be equipped with the [[Borel algebra]] generated by its open subsets unless otherwise specified. Most commonly this space will be the [[real numbers|real]] or [[complex numbers]]. For instance, a '''real-valued measurable function''' is a function for which the preimage of each [[Borel set]] is measurable. A '''complex-valued measurable function''' is defined analogously. In practice, some authors use '''measurable functions''' to refer only to real-valued measurable functions with respect to the Borel algebra.<ref name="strichartz">{{cite book | last = Strichartz | first = Robert | title = The Way of Analysis | publisher = Jones and Bartlett | year = 2000 | isbn = 0-7637-1497-6}}</ref> If the values of the function lie in an [[infinite-dimensional vector space]] instead of '''R''' or '''C''', usually other definitions of measurability are used, such as [[weak measurability]] and [[Bochner measurability]].
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| In [[probability theory]], the sigma algebra often represents the set of available information, and a function (in this context a [[random variable]]) is measurable if and only if it represents an outcome that is knowable based on the available information. In contrast, functions that are not Lebesgue measurable are generally considered [[Pathological (mathematics)|pathological]], at least in the field of [[mathematical analysis|analysis]].
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| ==Formal definition==
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| Let (''X'', Σ) and (''Y'', Τ) be measurable spaces, meaning that ''X'' and ''Y'' are sets equipped with respective sigma algebras Σ and Τ. A function ''f'': ''X'' → ''Y'' is said to be measurable if for every ''E'' ∈ Τ the preimage of ''E'' under ''f'' is in Σ; ie
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| :<math> f^{-1}(E) := \{ x\in X |\; f(x) \in E \} \in \Sigma,\;\; \forall E \in T. </math>
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| The notion of measurability depends on the sigma algebras Σ and Τ. To emphasize this dependency, if ''f'': ''X'' → ''Y'' is a measurable function, we will write
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| :<math> f \colon (X, \Sigma ) \rightarrow ( Y, T ) </math>
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| == Special measurable functions ==
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| * If (''X'', Σ) and (''Y'', Τ) are [[Borel space]]s, a measurable function ''f'': (''X'', Σ) → (''Y'', Τ) is also called a '''Borel function'''. [[Continuous function (topology)|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]]. If a Borel function happens to be a section of some map <math>Y\stackrel{\pi}{\to} X</math>, it is called a Borel section.
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| * A [[Lebesgue measurable]] function is a measurable function <math>f : (\mathbf{R}, \mathcal{L}) \to (\mathbf{C}, \mathcal{B}_\mathbf{C})</math>, where <math>\mathcal{L}</math> is the [[sigma algebra]] of [[Lebesgue measurable]] sets, and <math>\mathcal{B}_\mathbf{C}</math> is the [[Borel algebra]] on the [[complex number]]s '''C'''. Lebesgue measurable functions are of interest in [[mathematical analysis]] because they can be [[Lebesgue integration|integrated]].
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| * [[Random variable]]s are by definition measurable functions defined on [[sample space]]s.
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| == Properties of measurable functions ==
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| * The sum and product of two complex-valued measurable functions are measurable.<ref name="folland">{{cite book | last = Folland | first = Gerald B. | title = Real Analysis: Modern Techniques and their Applications | year = 1999 | publisher = Wiley | isbn = 0-471-31716-0}}</ref> So is the quotient, so long as there is no division by zero.<ref name="strichartz" />
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| * The composition of measurable functions is measurable; i.e., if ''f'': (''X'', Σ<sub>1</sub>) → (''Y'', Σ<sub>2</sub>) and ''g'': (''Y'', Σ<sub>2</sub>) → (''Z'', Σ<sub>3</sub>) are measurable functions, then so is ''g''(''f''(⋅)): (''X'', Σ<sub>1</sub>) → (''Z'', Σ<sub>3</sub>).<ref name="strichartz" /> But see the caveat regarding Lebesgue-measurable functions in the introduction.
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| * The (pointwise) [[supremum]], [[infimum]], [[limit superior]], and [[limit inferior]] of a sequence (viz., countably many) of real-valued measurable functions are all measurable as well.<ref name="strichartz" /><ref name="royden">{{cite book | last = Royden | first = H. L. | title = Real Analysis | year = 1988 | publisher = Prentice Hall | isbn = 0-02-404151-3 }}</ref>
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| *The [[pointwise]] limit of a sequence of measurable functions is measurable; note that the corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence. (This is correct when the counter domain of the elements of the sequence is a metric space. It is false in general; see pages 125 and 126 of.<ref name="dudley">{{cite book | last = Dudley | first = R. M. | title = Real Analysis and Probability | year = 2002 | edition = 2 | publisher = Cambridge University Press | isbn = 0-521-00754-2 }}</ref>)
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| ==Non-measurable functions==
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| Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to find non-measurable functions.
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| * So long as there are non-measurable sets in a measure space, there are non-measurable functions from that space. If (''X'', Σ) is some measurable space and ''A'' ⊂ ''X'' is a [[non-measurable set|non-measurable]] set, i.e. if ''A'' ∉ Σ, then the [[indicator function]] '''1'''<sub>''A''</sub>: (''X'', Σ) → '''R''' is non-measurable (where '''R''' is equipped with the [[Borel algebra]] as usual), since the preimage of the measurable set {1} is the non-measurable set ''A''. Here '''1'''<sub>''A''</sub> is given by
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| :<math>\mathbf{1}_A(x) = \begin{cases}
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| 1 & \text{ if } x \in A \\
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| 0 & \text{ otherwise}
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| \end{cases}</math>
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| * Any non-constant function can be made non-measurable by equipping the domain and range with appropriate σ-algebras. If ''f'': ''X'' → '''R''' is an arbitrary non-constant, real-valued function, then ''f'' is non-measurable if ''X'' is equipped with the indiscrete algebra Σ = {0, ''X''}, since the preimage of any point in the range is some proper, nonempty subset of ''X'', and therefore does not lie in Σ.
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| ==See also==
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| *Vector spaces of measurable functions: the [[Lp space|''L<sup>p</sup>'' spaces]]
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| *[[Measure-preserving dynamical system]]
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| ==Notes==
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| {{Reflist}}
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| ==External links==
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| * [http://www.encyclopediaofmath.org/index.php/Measurable_function Measurable function] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| * [http://www.encyclopediaofmath.org/index.php/Borel_function Borel function] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| {{DEFAULTSORT:Measurable Function}}
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| [[Category:Measure theory]]
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| [[Category:Types of functions]]
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Hello. Let me introduce the writer. Her name is Refugia Shryock. Years in the past he moved to North Dakota and his family members enjoys it. One of the issues she loves most is to read comics and she'll be beginning some thing else alongside with it. I am a meter reader but I strategy on changing it.
Here is my site - std home test