imported>Trovatore: shorten short desc -- doesn't need to be a technical definition; just needs to give broad context

2024-11-09T22:12:53Z

shorten short desc -- doesn't need to be a technical definition; just needs to give broad context

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{{Short description|Kind of mathematical function}}
{{Use American English|date = January 2019}}

In [[mathematics]], and in particular [[Mathematical analysis#Measure_theory|measure theory]], a '''measurable function''' is a function between the underlying sets of two [[measurable space|measurable spaces]] that preserves the structure of the spaces: the [[preimage]] of any [[Measure (mathematics)|measurable]] set is measurable. This is in direct analogy to the definition that a [[Continuous function|continuous]] function between [[topological space|topological spaces]] [[Morphism|preserves]] the topological structure: the preimage of any [[open set]] is open. In [[real analysis]], measurable functions are used in the definition of the [[Lebesgue integration|Lebesgue integral]]. In [[probability theory]], a measurable function on a [[probability space]] is known as a [[random variable]].

== Formal definition ==

Let <math>(X,\Sigma)</math> and <math>(Y,\Tau)</math> be measurable spaces, meaning that <math>X</math> and <math>Y</Math> are sets equipped with respective [[σ-algebra|<math>\sigma</math>-algebras]] <math>\Sigma</math> and <math>\Tau.</math> A function <math>f:X\to Y</math> is said to be measurable if for every <math>E\in \Tau</math> the pre-image of <math>E</math> under <math>f</math> is in <math>\Sigma</math>; that is, for all <math>E \in \Tau </math>
<math display="block">f^{-1}(E) := \{ x\in X \mid f(x) \in E \} \in \Sigma.</math>

That is, <math>\sigma (f)\subseteq\Sigma,</math> where <math>\sigma (f)</math> is the [[Σ-algebra#σ-algebra_generated_by_a_function|σ-algebra generated by f]]. If <math>f:X\to Y</math> is a measurable function, one writes
<math display="block">f \colon (X, \Sigma) \rightarrow (Y, \Tau).</math>
to emphasize the dependency on the <math>\sigma</math>-algebras <math>\Sigma</math> and <math>\Tau.</math>

== Term usage variations ==

The choice of <math>\sigma</math>-algebras in the definition above is sometimes implicit and left up to the context. For example, for <math>\R,</math> <math>\Complex,</math> or other topological spaces, the [[Borel algebra]] (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.<ref name="strichartz">{{cite book|last=Strichartz|first=Robert|title=The Way of Analysis|url=https://archive.org/details/wayofanalysis0000stri|url-access=registration|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]], 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 <math>(X, \Sigma)</math> and <math>(Y, T)</math> are [[Borel set#Standard Borel spaces and Kuratowski theorems|Borel space]]s, a measurable function <math>f:(X, \Sigma) \to (Y, T)</math> 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]]. If a Borel function happens to be a section of a map <math>Y\xrightarrow{~\pi~}X,</math> it is called a '''Borel section'''.
* A [[Lebesgue measurable]] function is a measurable function <math>f : (\R, \mathcal{L}) \to (\Complex, \mathcal{B}_\Complex),</math> where <math>\mathcal{L}</math> is the <math>\sigma</math>-algebra of Lebesgue measurable sets, and <math>\mathcal{B}_\Complex</math> is the [[Borel algebra]] on the [[complex number]]s <math>\Complex.</math> Lebesgue measurable functions are of interest in [[mathematical analysis]] because they can be integrated. In the case <math>f : X \to \R,</math> <math>f</math> is Lebesgue measurable if and only if <math>\{f > \alpha\} = \{ x\in X : f(x) > \alpha\}</math> is measurable for all <math>\alpha\in\R.</math> This is also equivalent to any of <math>\{f \geq \alpha\},\{f<\alpha\},\{f\le\alpha\}</math> being measurable for all <math>\alpha,</math> 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.<ref name="carothers">{{cite book |last=Carothers|first=N. L.|title=Real Analysis|url=https://archive.org/details/realanalysis0000caro| url-access=registration | year=2000| publisher=Cambridge University Press| isbn=0-521-49756-6}}</ref> A function <math>f:X\to\Complex</math> 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.<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" />
* If <math>f : (X,\Sigma_1) \to (Y,\Sigma_2)</math> and <math>g:(Y,\Sigma_2) \to (Z,\Sigma_3)</math> are measurable functions, then so is their composition <math>g\circ f:(X,\Sigma_1) \to (Z,\Sigma_3).</math><ref name="strichartz" />
* If <math>f : (X,\Sigma_1) \to (Y,\Sigma_2)</math> and <math>g:(Y,\Sigma_3) \to (Z,\Sigma_4)</math> are measurable functions, their composition <math>g\circ f: X\to Z</math> need not be <math>(\Sigma_1,\Sigma_4)</math>-measurable unless <math>\Sigma_3 \subseteq \Sigma_2.</math> Indeed, two Lebesgue-measurable functions may be constructed in such a way as to make their composition non-Lebesgue-measurable.
* 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>
*The [[pointwise]] limit of a sequence of measurable functions <math>f_n: X \to Y</math> is measurable, where <math>Y</math> is a metric space (endowed with the Borel algebra). This is not true in general if <math>Y</math> is non-metrizable. The corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence.<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><ref name="aliprantis">{{cite book|last1=Aliprantis|first1=Charalambos D.|last2=Border|first2=Kim C.|title=Infinite Dimensional Analysis, A Hitchhiker's Guide|year=2006|edition=3|publisher=Springer|isbn=978-3-540-29587-7}}</ref>

== 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 an essential way, in the sense that [[Zermelo–Fraenkel set theory]] without the axiom of choice does not prove the existence of such functions.

In any measure space ''<math>(X, \Sigma)</math>'' with a [[non-measurable set]] <math>A \subset X,</math> <math>A \notin \Sigma,</math> one can construct a non-measurable [[indicator function]]:
<math display="block">\mathbf{1}_A:(X,\Sigma) \to \R,
\quad
\mathbf{1}_A(x) = \begin{cases}
1 & \text{ if } x \in A \\
0 & \text{ otherwise},
\end{cases}</math>
where <math>\R</math> is equipped with the usual [[Borel algebra]]. This is a non-measurable function since the preimage of the measurable set <math>\{1\}</math> is the non-measurable <math>A.</math>

As another example, any non-constant function <math>f : X \to \R</math> is non-measurable with respect to the trivial <math>\sigma</math>-algebra <math>\Sigma = \{\varnothing, X\},</math> since the preimage of any point in the range is some proper, nonempty subset of <math>X,</math> which is not an element of the trivial <math>\Sigma.</math>

== See also ==

* {{annotated link|Bochner measurable function}}
* {{annotated link|Bochner space}}
* {{annotated link|Lp space}} - Vector spaces of measurable functions: the [[Lp space|<math>L^p</math> spaces]]
* {{annotated link|Measure-preserving dynamical system}}
* {{annotated link|Vector measure}}
* {{annotated link|Weakly measurable function}}

== Notes ==

{{reflist|group=note}}
{{reflist}}

==External links==

* [http://www.encyclopediaofmath.org/index.php/Measurable_function Measurable function] at [[Encyclopedia of Mathematics]]
* [http://www.encyclopediaofmath.org/index.php/Borel_function Borel function] at [[Encyclopedia of Mathematics]]

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imported>Trovatore: shorten short desc -- doesn't need to be a technical definition; just needs to give broad context