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In `F33f`_`[mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics]`_`f, and in particular `F33f`_`[measure theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematical_analysis]`_`f, a `!measurable function`! is a function between the underlying sets of two `F33f`_`[measurable spaces`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Measurable_space]`_`f that preserves the structure of the spaces: the `F33f`_`[preimage`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Preimage]`_`f of any `F33f`_`[measurable`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Measure_(mathematics)]`_`f set is measurable. This is in direct analogy to the definition that a `F33f`_`[continuous`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Continuous_function]`_`f function between `F33f`_`[topological spaces`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Topological_space]`_`f `F33f`_`[preserves`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Morphism]`_`f the topological structure: the preimage of any `F33f`_`[open set`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Open_set]`_`f is open. In `F33f`_`[real analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Real_analysis]`_`f, measurable functions are used in the definition of the `F33f`_`[Lebesgue integral`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lebesgue_integration]`_`f. In `F33f`_`[probability theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Probability_theory]`_`f, a measurable function on a `F33f`_`[probability space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Probability_space]`_`f is known as a `F33f`_`[random variable`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Random_variable]`_`f.

>>Contents

• `F0af`_`[Formal definition`#formal-definition]`_`f

• `F0af`_`[Term usage variations`#term-usage-variations]`_`f

• `F0af`_`[Notable classes of measurable functions`#notable-classes-of-measurable-functions]`_`f

• `F0af`_`[Properties of measurable functions`#properties-of-measurable-functions]`_`f

• `F0af`_`[Non-measurable functions`#non-measurable-functions]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[Notes`#notes]`_`f

• `F0af`_`[External links`#external-links]`_`f

-─

>>Formal definition

Let ( X , Σ 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 `F33f`_`[σ {\displaystyle \sigma } -algebras`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Σ-algebra]`_`f Σ Σ {\\displaystyle \\Sigma } and T . {\\displaystyle \\mathrm {T} .} A function f : X → → Y {\\displaystyle f:X\\to Y} is said to be measurable if for every E ∈ ∈ T {\\displaystyle E\\in \\mathrm {T} } the pre-image of E {\\displaystyle E} under f {\\displaystyle f} is in Σ Σ {\\displaystyle \\Sigma } ; that is, for all E ∈ ∈ T {\\displaystyle E\\in \\mathrm {T} } f − − 1 ( E ) := { x ∈ ∈ X ∣ ∣ f ( x ) ∈ ∈ E } ∈ ∈ Σ Σ . {\\displaystyle f^{-1}(E):=\\{x\\in X\\mid f(x)\\in E\\}\\in \\Sigma .}

That is, σ σ ( f ) ⊆ ⊆ Σ Σ , {\\displaystyle \\sigma (f)\\subseteq \\Sigma ,} where σ σ ( f ) {\\displaystyle \\sigma (f)} is the `F33f`_`[σ-algebra generated by f`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Σ-algebra]`_`f. If f : X → → Y {\\displaystyle f:X\\to Y} is a measurable function, one writes f : : ( X , Σ Σ ) → → ( Y , T ) . {\\displaystyle f\\colon (X,\\Sigma )\\rightarrow (Y,\\mathrm {T} ).} to emphasize the dependency on the σ σ {\\displaystyle \\sigma } -algebras Σ Σ {\\displaystyle \\Sigma } and T . {\\displaystyle \\mathrm {T} .}

>>Term usage variations

The choice of σ 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 spaces, the `F33f`_`[Borel algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Borel_algebra]`_`f (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.`:cite-ref-strichartz-1-0[`F5bf`_`[1`#cite-note-strichartz-1]`_`f]

If the values of the function lie in an `F33f`_`[infinite-dimensional vector space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Infinite-dimensional_vector_space]`_`f, other non-equivalent definitions of measurability, such as `F33f`_`[weak measurability`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Weak_measurability]`_`f and `F33f`_`[Bochner measurability`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bochner_measurability]`_`f, exist.

>>Notable classes of measurable functions

• Random variables are by definition measurable functions defined on probability spaces.

• If ( X , Σ Σ ) {\\displaystyle (X,\\Sigma )} and ( Y , T ) {\\displaystyle (Y,T)} are `F33f`_`[Borel spaces`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Borel_set]`_`f, a measurable function f : ( X , Σ 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 `F33f`_`[Luzin's theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Luzin's_theorem]`_`f. If a Borel function happens to be a section of a map Y → π π X , {\\displaystyle Y\\xrightarrow {~\\pi ~} X,} it is called a `!Borel section`!.

• A `F33f`_`[Lebesgue measurable`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lebesgue_measurable]`_`f 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 σ {\\displaystyle \\sigma } -algebra of Lebesgue measurable sets, and B C {\\displaystyle {\\mathcal {B}}_{\\mathbb {C} }} is the `F33f`_`[Borel algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Borel_algebra]`_`f on the `F33f`_`[complex numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_number]`_`f C . {\\displaystyle \\mathbb {C} .} Lebesgue measurable functions are of interest in `F33f`_`[mathematical analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematical_analysis]`_`f because they can be integrated. In the case f : X → → R , {\\displaystyle f:X\\to \\mathbb {R} ,} f {\\displaystyle f} is Lebesgue measurable if and only if { f > α α } = { x ∈ ∈ X : f ( x ) > α α } {\\displaystyle \\{f>\\alpha \\}=\\{x\\in X:f(x)>\\alpha \\}} is measurable for all α α ∈ ∈ R . {\\displaystyle \\alpha \\in \\mathbb {R} .} This is also equivalent to any of { f ≥ ≥ α α } , { f < α α } , { f ≤ ≤ α α } {\\displaystyle \\{f\\geq \\alpha \\},\\{f<\\alpha \\},\\{f\\leq \\alpha \\}} being measurable for all α α , {\\displaystyle \\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.`:cite-ref-carothers-2-0[`F5bf`_`[2`#cite-note-carothers-2]`_`f] A function f : X → → C {\\displaystyle f:X\\to \\mathbb {C} } 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.`:cite-ref-folland-3-0[`F5bf`_`[3`#cite-note-folland-3]`_`f] So is the quotient, so long as there is no division by zero.`:cite-ref-strichartz-1-1[`F5bf`_`[1`#cite-note-strichartz-1]`_`f]

• If f : ( X , Σ Σ 1 ) → → ( Y , Σ Σ 2 ) {\\displaystyle f:(X,\\Sigma _{1})\\to (Y,\\Sigma _{2})} and g : ( Y , Σ Σ 2 ) → → ( Z , Σ Σ 3 ) {\\displaystyle g:(Y,\\Sigma _{2})\\to (Z,\\Sigma _{3})} are measurable functions, then so is their composition g ∘ ∘ f : ( X , Σ Σ 1 ) → → ( Z , Σ Σ 3 ) . {\\displaystyle g\\circ f:(X,\\Sigma _{1})\\to (Z,\\Sigma _{3}).} `:cite-ref-strichartz-1-2[`F5bf`_`[1`#cite-note-strichartz-1]`_`f]

• If f : ( X , Σ Σ 1 ) → → ( Y , Σ Σ 2 ) {\\displaystyle f:(X,\\Sigma _{1})\\to (Y,\\Sigma _{2})} and g : ( Y , Σ Σ 3 ) → → ( Z , Σ 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 Σ 1 , Σ Σ 4 ) {\\displaystyle (\\Sigma _{1},\\Sigma _{4})} -measurable unless Σ Σ 3 ⊆ ⊆ Σ Σ 2 . {\\displaystyle \\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) `F33f`_`[supremum`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Supremum]`_`f, `F33f`_`[infimum`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Infimum]`_`f, `F33f`_`[limit superior`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Limit_superior]`_`f, and `F33f`_`[limit inferior`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Limit_inferior]`_`f of a sequence (viz., countably many) of real-valued measurable functions are all measurable as well.`:cite-ref-strichartz-1-3[`F5bf`_`[1`#cite-note-strichartz-1]`_`f]`:cite-ref-royden-4-0[`F5bf`_`[4`#cite-note-royden-4]`_`f]

• The `F33f`_`[pointwise`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pointwise]`_`f 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. The corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence.`:cite-ref-dudley-5-0[`F5bf`_`[5`#cite-note-dudley-5]`_`f]`:cite-ref-aliprantis-6-0[`F5bf`_`[6`#cite-note-aliprantis-6]`_`f]

>>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 `F33f`_`[axiom of choice`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Axiom_of_choice]`_`f in an essential way, in the sense that `F33f`_`[Zermelo–Fraenkel set theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Zermelo–Fraenkel_set_theory]`_`f without the axiom of choice does not prove the existence of such functions.

In any measure space `* ( X , Σ Σ ) {\\displaystyle (X,\\Sigma )} `* with a `F33f`_`[non-measurable set`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Non-measurable_set]`_`f A ⊂ ⊂ X , {\\displaystyle A\\subset X,} A ∉ ∉ Σ Σ , {\\displaystyle A\\notin \\Sigma ,} one can construct a non-measurable `F33f`_`[indicator function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Indicator_function]`_`f: 1 A : ( X , Σ Σ ) → → R , 1 A ( x ) = { 1 if x ∈ ∈ A 0 otherwise , {\\displaystyle \\mathbf {1} _{A}:(X,\\Sigma )\\to \\mathbb {R} ,\\quad \\mathbf {1} _{A}(x)={\\begin{cases}1&{\\text{ if }}x\\in A\\\\0&{\\text{ otherwise}},\\end{cases}}} where R {\\displaystyle \\mathbb {R} } is equipped with the usual `F33f`_`[Borel algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Borel_algebra]`_`f. This is a non-measurable function since the preimage of the measurable set { 1 } {\\displaystyle \\{1\\}} is the non-measurable A . {\\displaystyle A.}

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

>>See also

• `F33f`_`[Bochner measurable function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bochner_measurable_function]`_`f

• `F33f`_`[Bochner space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bochner_space]`_`f – Type of topological space

• `F33f`_`[Lp space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lp_space]`_`f – Function spaces generalizing finite-dimensional p norm spaces - Vector spaces of measurable functions: the `F33f`_`[L p {\displaystyle L^{p}} spaces`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lp_space]`_`f

• `F33f`_`[Measure-preserving dynamical system`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Measure-preserving_dynamical_system]`_`f – Subject of study in ergodic theory

• `F33f`_`[Vector measure`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Vector_measure]`_`f

• `F33f`_`[Weakly measurable function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Weakly_measurable_function]`_`f

>>Notes

`:cite-note-strichartz-1`!1.`! `F0af`_`[↑`#cite-ref-strichartz-1-0]`_`f `:citerefstrichartz2000`aStrichartz, Robert (2000). `*The Way of Analysis`*. Jones and Bartlett. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-7637-1497-6.

`:cite-note-carothers-2`!2.`! `F0af`_`[↑`#cite-ref-carothers-2-0]`_`f `:citerefcarothers2000`aCarothers, N. L. (2000). `*Real Analysis`*. Cambridge University Press. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-521-49756-6.

`:cite-note-folland-3`!3.`! `F0af`_`[↑`#cite-ref-folland-3-0]`_`f `:citereffolland1999`aFolland, Gerald B. (1999). `*Real Analysis: Modern Techniques and their Applications`*. Wiley. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-471-31716-0.

`:cite-note-royden-4`!4.`! `F0af`_`[↑`#cite-ref-royden-4-0]`_`f `:citerefroyden1988`aRoyden, H. L. (1988). `*Real Analysis`*. Prentice Hall. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-02-404151-3.

`:cite-note-dudley-5`!5.`! `F0af`_`[↑`#cite-ref-dudley-5-0]`_`f `:citerefdudley2002`aDudley, R. M. (2002). `*Real Analysis and Probability`* (2 ed.). Cambridge University Press. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-521-00754-2.

`:cite-note-aliprantis-6`!6.`! `F0af`_`[↑`#cite-ref-aliprantis-6-0]`_`f `:citerefaliprantisborder2006`aAliprantis, Charalambos D.; Border, Kim C. (2006). `*Infinite Dimensional Analysis, A Hitchhiker's Guide`* (3 ed.). Springer. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-3-540-29587-7.

>>External links

• Measurable function at `F33f`_`[Encyclopedia of Mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Encyclopedia_of_Mathematics]`_`f

• Borel function at `F33f`_`[Encyclopedia of Mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Encyclopedia_of_Mathematics]`_`f

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