Measurable function - wikipedia Measurable function In mathematics and in particular measure theory, a measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable, analogously to the definition that a function between topological spaces is continuous if the preimage of each open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable. Contents 1 Formal definition 2 Term usage variations 3 Notable classes of measurable functions 4 Properties of measurable functions 5 Non-measurable functions 6 See also 7 Notes 8 External links Formal definition Let ( X, Σ ) ( displaystyle ( X, Sigma ) ) and ( Y, T ) ( displaystyle ( Y, mathrm ( T ) ) ) be measurable spaces, meaning that X ( displaystyle X ) and Y ( displaystyle Y ) are sets equipped with respective σ ( displaystyle sigma ) - algebras Σ ( displaystyle Sigma ) and T ( displaystyle mathrm ( T ) ). A function f : X → Y ( displaystyle f : X \to Y ) is said to be measurable if the preimage of E ( displaystyle E ) under f ( displaystyle f ) is in Σ ( displaystyle Sigma ) for every E ∈ T ( displaystyle E in mathrm ( T ) ) ; i.e. f − 1 ( E ) : = ( x ∈ X f ( x ) ∈ E ) ∈ Σ, ∀ E ∈ T. ( displaystyle f ^ ( - 1 ) ( E ) : = ( x in X ; f ( x ) in E ) in Sigma, ; ; \forall E in mathrm ( T ). ) If f : X → Y ( displaystyle f : X \to Y ) is a measurable function, we will write f : ( X, Σ ) → ( Y, T ). ( displaystyle f colon ( X, Sigma ) \rightarrow ( Y, mathrm ( T ) ). ) to emphasize the dependency on the σ ( displaystyle sigma ) - algebras Σ ( displaystyle Sigma ) and T ( displaystyle mathrm ( T ) ). Term usage variations The choice of σ ( displaystyle sigma ) - algebras in the definition above is sometimes implicit and left up to the context. For example, for R ( displaystyle ( mathbb ( R ) ) ), C ( displaystyle ( mathbb ( C ) ) ), or other topological space, the Borel algebra ( containing 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, 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, Σ ) ( displaystyle ( X, Sigma ) ) and ( Y, T ) ( displaystyle ( Y, T ) ) are Borel spaces, a measurable function f : ( X, Σ ) → ( Y, T ) ( displaystyle 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. If a Borel function happens to be a section of some map Y → π X ( displaystyle Y ( \xrightarrow ( ~ pi ~ ) ) X ), it is called a Borel section. A Lebesgue measurable function is a measurable function f : ( R, L ) → ( C, B C ) ( displaystyle f : ( mathbb ( R ), ( mathcal ( L ) ) ) \to ( mathbb ( C ), ( mathcal ( B ) ) _ ( mathbb ( C ) ) ) ), where L ( displaystyle ( mathcal ( L ) ) ) is the σ ( displaystyle sigma ) - algebra of Lebesgue measurable sets, and B C ( displaystyle ( mathcal ( B ) ) _ ( mathbb ( C ) ) ) is the Borel algebra on the complex numbers C ( displaystyle mathbb ( C ) ). Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated. In the case f : X → R ( displaystyle f : X \to mathbb ( R ) ), f ( displaystyle f ) is Lebesgue measurable iff ( f > α ) = ( x ∈ X : f ( x ) > α ) ( displaystyle ( f > alpha ) = ( x in X : f ( x ) > alpha ) ) is measurable for all α ∈ R ( displaystyle alpha in mathbb ( R ) ). This is also equivalent to any of ( f ≥ α ), ( f < α ), ( f ≤ α ) ( displaystyle ( f geq alpha ), ( f < alpha ), ( f leq alpha ) ) being measurable for all α ( displaystyle alpha ). Continuous functions, monotone functions, step functions, semicontinuous functions, Riemann - integrable functions, and functions of bounded variation are all Lebesgue measurable. A function f : X → C ( displaystyle f : X \to mathbb ( C ) ) is measurable iff 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, Σ 1 ) → ( Y, Σ 2 ) ( displaystyle f : ( X, Sigma _ ( 1 ) ) \to ( Y, Sigma _ ( 2 ) ) ) and g : ( Y, Σ 2 ) → ( Z, Σ 3 ) ( displaystyle g : ( Y, Sigma _ ( 2 ) ) \to ( Z, Sigma _ ( 3 ) ) ) are measurable functions, then so is their composition g ∘ f : ( X, Σ 1 ) → ( Z, Σ 3 ) ( displaystyle g circ f : ( X, Sigma _ ( 1 ) ) \to ( Z, Sigma _ ( 3 ) ) ). If f : ( X, Σ 1 ) → ( Y, Σ 2 ) ( displaystyle f : ( X, Sigma _ ( 1 ) ) \to ( Y, Sigma _ ( 2 ) ) ) and g : ( Y, Σ 3 ) → ( Z, Σ 4 ) ( displaystyle g : ( Y, Sigma _ ( 3 ) ) \to ( Z, Sigma _ ( 4 ) ) ) are measurable functions, their composition g ∘ f : X → Z ( displaystyle g circ f : X \to Z ) need not be ( Σ 1, Σ 4 ) ( displaystyle ( Sigma _ ( 1 ), Sigma _ ( 4 ) ) ) - measurable unless Σ 2 ( displaystyle Sigma _ ( 2 ) ) and Σ 3 ( displaystyle Sigma _ ( 3 ) ) are the same. 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. The pointwise limit of a sequence of measurable functions f n : X → Y ( displaystyle f_ ( n ) : X \to Y ) is measurable, where Y ( displaystyle Y ) is a metric space ( endowed with the Borel algebra ). This is not true in general if Y ( displaystyle 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 find non-measurable functions. So long as there are non-measurable sets in a measure space, there are non-measurable functions from that space. If ( X, Σ ) ( displaystyle ( X, Sigma ) ) is some measurable space and A ⊂ X ( displaystyle A subset X ) is a non-measurable set, i.e. if A ∉ Σ ( displaystyle A \notin Sigma ), then the indicator function 1 A : ( X, Σ ) → R ( displaystyle mathbf ( 1 ) _ ( A ) : ( X, Sigma ) \to mathbb ( R ) ) is non-measurable ( where R ( displaystyle mathbb ( R ) ) is equipped with the Borel algebra as usual ), since the preimage of the measurable set ( 1 ) ( displaystyle ( 1 ) ) is the non-measurable set A ( displaystyle A ). Here 1 A ( displaystyle mathbf ( 1 ) _ ( A ) ) is given by 1 A ( x ) = ( 1 if x ∈ A ; 0 otherwise. ( displaystyle mathbf ( 1 ) _ ( A ) ( x ) = ( \begin ( cases ) 1& ( \text ( if ) ) x in A ; 0& ( \text ( otherwise ) ). end ( cases ) ) ) Any non-constant function can be made non-measurable by equipping the domain and range with appropriate σ ( displaystyle sigma ) - algebras. If f : X → R ( displaystyle f : X \to mathbb ( R ) ) is an arbitrary non-constant, real - valued function, then f ( displaystyle f ) is non-measurable if X ( displaystyle X ) is equipped with the trivial σ ( displaystyle sigma ) - algebra Σ = ( ∅, X ) ( displaystyle Sigma = ( emptyset, X ) ), since the preimage of any point in the range is some proper, nonempty subset of X ( displaystyle X ), and therefore does not lie in Σ ( displaystyle Sigma ). See also Vector spaces of measurable functions : the L p ( displaystyle L ^ ( p ) ) spaces Measure - preserving dynamical system Notes ^ Jump up to : Strichartz, Robert ( 2000 ). The Way of Analysis. Jones and Bartlett. ISBN 0 - 7637 - 1497 - 6. Jump up ^ Carothers, N.L. ( 2000 ). Real Analysis. Cambridge University Press. ISBN 0 - 521 - 49756 - 6. Jump up ^ Folland, Gerald B. ( 1999 ). Real Analysis : Modern Techniques and their Applications. Wiley. ISBN 0 - 471 - 31716 - 0. Jump up ^ Royden, H.L. ( 1988 ). Real Analysis. Prentice Hall. ISBN 0 - 02 - 404151 - 3. Jump up ^ Dudley, R.M. ( 2002 ). Real Analysis and Probability ( 2 ed. ). Cambridge University Press. ISBN 0 - 521 - 00754 - 2. Jump up ^ Aliprantis, Charalambos D. ; Border, Kim C. ( 2006 ). Infinite Dimensional Analysis, A Hitchhiker 's Guide ( 3 ed. ). Springer. ISBN 978 - 3 - 540 - 29587 - 7. External links Measurable function at Encyclopedia of Mathematics Borel function at Encyclopedia of Mathematics Retrieved from `` https://en.wikipedia.org/w/index.php?title=Measurable_function&oldid=794787021 '' Categories : Measure theory Types of functions Talk Contents About Wikipedia Català Čeština Deutsch Español Français 한국어 Italiano עברית Қазақша Nederlands 日本 語 Polski Português Русский Svenska Українська 中文 Edit links This page was last edited on 10 August 2017, at 01 : 44. Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. 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Question: when is a function said to be measurable
Answer: a function between two measurable spaces such that the preimage of any measurable set is measurable