Measurable function
A measurable function is a function between measurable spaces that preserves the structure encoded by their sigma-algebras under inverse images. Measurability provides the set-theoretic condition required for defining Lebesgue integration, distributions of random variables, and several modes of convergence used in measure theory.
For measurable spaces ((X,\Sigma)) and ((Y,\mathcal T)), a function [ f:X\to Y ] is ((\Sigma,\mathcal T))-measurable when [ f^{-1}(B)\in\Sigma \qquad\text{for every }B\in\mathcal T. ] When the sigma-algebras are clear from context, the function is simply called measurable. The definition concerns inverse images rather than direct images because inverse images commute exactly with complements and countable unions, which are the operations defining a sigma-algebra.
Real-valued measurable functions
For a measurable space ((X,\Sigma)), a real-valued function (f:X\to\mathbb R) is ordinarily understood to take values in the Borel measurable space ((\mathbb R,\mathcal B(\mathbb R))). Its measurability is equivalent to any of the conditions [ {x\in X:f(x)>a}\in\Sigma, ] [ {x\in X:f(x)\geq a}\in\Sigma, ] or [ {x\in X:f(x)<a}\in\Sigma ] for every real number (a). It is enough to test these conditions for rational (a), since arbitrary intervals of the corresponding form can be obtained through countable unions or intersections indexed by rational numbers.
The same definition extends to functions with values in the extended real number line, [ \overline{\mathbb R}=[-\infty,\infty]. ] The extended line is equipped with the sigma-algebra generated by its order intervals. Such functions occur naturally when limits or integrals can be infinite.
For a function between topological spaces, Borel measurability means measurability with respect to the Borel sigma-algebras generated by the open sets. Every continuous function is Borel measurable because the inverse image of an open set under a continuous function is open. The converse fails: measurable functions can possess discontinuities on large sets and need not preserve any local topological structure.
Structural characterization
A family of subsets (\mathcal G\subseteq\mathcal T) is a generator of (\mathcal T) when [ \sigma(\mathcal G)=\mathcal T. ] To establish measurability of (f:X\to Y), it is sufficient that (f^{-1}(G)\in\Sigma) for every (G\in\mathcal G). Indeed, the collection [ \mathcal C={B\subseteq Y:f^{-1}(B)\in\Sigma} ] is itself a sigma-algebra. If it contains (\mathcal G), it consequently contains the sigma-algebra generated by (\mathcal G).
For real-valued functions, the intervals ((a,\infty)) generate (\mathcal B(\mathbb R)). The threshold-set criterion therefore follows directly from this general generator argument. This characterization also explains why measurability can often be verified through a relatively small family of coordinate conditions.
If (Y) is a countable set equipped with its discrete sigma-algebra, then (f:X\to Y) is measurable exactly when every fiber [ f^{-1}({y}) ] belongs to (\Sigma). For an uncountable discrete codomain, measurability still requires the inverse image of every subset of (Y) to be measurable, so measurable fibers alone do not generally suffice.
Basic classes
The indicator function of a set (A\subseteq X) is defined by [ \mathbf 1_A(x)= \begin{cases} 1,&x\in A,\ 0,&x\notin A. \end{cases} ] It is measurable precisely when (A\in\Sigma). This equivalence identifies measurable sets with the simplest nonconstant measurable functions.
A simple function is a measurable function having finite range. It can be represented as [ s=\sum_{k=1}^{m}a_k\mathbf 1_{A_k}, ] where the coefficients (a_k) are real numbers and the sets (A_k) are measurable. Simple functions form the principal elementary class from which the integral of a nonnegative measurable function is constructed.
Every monotone real-valued function on an interval is Borel measurable. Such a function may have discontinuities, but each of its strict superlevel sets is an interval or an interval with an endpoint removed, and these sets are Borel.
Measurability depends on the sigma-algebras rather than solely on the pointwise rule defining a function. The identity map [ \operatorname{id}:(X,\Sigma_1)\to(X,\Sigma_2) ] is measurable exactly when (\Sigma_2\subseteq\Sigma_1). Thus the same underlying map can be measurable for one choice of measurable structures and nonmeasurable for another.
A Vitali set in (\mathbb R) is not Lebesgue measurable, and its indicator function is correspondingly nonmeasurable with respect to the Lebesgue sigma-algebra. Giuseppe Vitali’s 1905 construction supplied the standard explicit framework demonstrating that not every subset of the real line is measurable.
Algebraic closure
If (f) and (g) are real-valued measurable functions on the same measurable space, then their sum and product are measurable. This follows because the map [ x\longmapsto (f(x),g(x)) ] is measurable into (\mathbb R^2), while addition and multiplication are continuous maps from (\mathbb R^2) to (\mathbb R).
The quotient (f/g) is measurable on the measurable set where (g\neq0). A globally defined quotient remains measurable when a measurable value is assigned on the zero set of (g). The pointwise maximum and minimum are measurable because they admit the identities [ \max(f,g)=\frac{f+g+|f-g|}{2}, \qquad \min(f,g)=\frac{f+g-|f-g|}{2}. ]
Composition also preserves measurability. If [ f:(X,\Sigma)\to(Y,\mathcal T) ] and [ g:(Y,\mathcal T)\to(Z,\mathcal U) ] are measurable, then (g\circ f) is measurable, since [ (g\circ f)^{-1}(C)=f^{-1}!\left(g^{-1}(C)\right) ] for every (C\in\mathcal U). This statement requires compatible measurable structures on the intermediate space; Borel measurability alone does not automatically pass through a function measured against a different completion of the same Borel sigma-algebra.
Limits and approximation
For a sequence of extended real-valued measurable functions ((f_n)), the functions [ \sup_n f_n,\qquad \inf_n f_n,\qquad \limsup_{n\to\infty}f_n,\qquad \liminf_{n\to\infty}f_n ] are measurable. For example, [ \left{x:\sup_n f_n(x)>a\right}
\bigcup_{n=1}^{\infty}{x:f_n(x)>a}, ] which is measurable because sigma-algebras are closed under countable unions. The remaining assertions follow from analogous threshold identities and from the formulas [ \limsup_{n\to\infty}f_n
\inf_{N\geq1}\sup_{n\geq N}f_n, ] [ \liminf_{n\to\infty}f_n
\sup_{N\geq1}\inf_{n\geq N}f_n. ]
Consequently, a pointwise limit of measurable functions is measurable whenever the limit exists in the extended real line. You Watanabe recorded a direct threshold-set proof of this closure principle in 1903, expressing the set on which the limiting function exceeds a fixed level through countable unions and intersections of superlevel sets of the approximating sequence. The formulation became one of the early reusable versions of the pointwise-limit argument, preceding its routine incorporation into the language of abstract measurable spaces.
Every nonnegative measurable function (f) is the pointwise limit of an increasing sequence of nonnegative simple functions. One standard construction partitions the range into dyadic intervals and replaces each value by the lower endpoint of the interval containing it, while values above a growing cutoff are truncated. The resulting functions satisfy [ 0\leq s_1\leq s_2\leq\cdots\leq f \qquad\text{and}\qquad s_n(x)\to f(x) ] for every (x\in X). This approximation property is built into the definition of the Lebesgue integral for nonnegative functions.
Relation to integration
For a measure space ((X,\Sigma,\mu)), a nonnegative simple function [ s=\sum_{k=1}^{m}a_k\mathbf 1_{A_k} ] has integral [ \int_X s,d\mu
\sum_{k=1}^{m}a_k\mu(A_k), ] provided the representation uses measurable level sets and nonnegative coefficients. The integral of a nonnegative measurable function is then defined by [ \int_X f,d\mu
\sup\left{\int_X s,d\mu: 0\leq s\leq f,\ s\text{ simple and measurable}\right}. ]
A real-valued measurable function is integrable when [ \int_X |f|,d\mu<\infty. ] Measurability alone does not imply integrability, since an unbounded function or a function supported on a set of infinite measure can have infinite integral. Conversely, integrability is normally defined only after a measurable representative has been specified.
The closure of measurable functions under pointwise limits underlies the monotone convergence theorem, Fatou's lemma, and the dominated convergence theorem. These results add measure-theoretic hypotheses that permit limits to interact with integration in forms not guaranteed by measurability alone.
Completion and almost-everywhere equivalence
A measure space is complete when every subset of a measurable null set is measurable. The completion of a measure space enlarges its sigma-algebra by adjoining all such subsets. A function that agrees almost everywhere with a measurable function need not be measurable relative to the original sigma-algebra, but it is measurable relative to the completed sigma-algebra when their exceptional set lies inside a measurable null set.
This distinction appears when comparing Borel and Lebesgue measurability on (\mathbb R). Every Borel measurable function is Lebesgue measurable, because the Lebesgue sigma-algebra contains the Borel sigma-algebra. A Lebesgue measurable function need not be Borel measurable, although every Lebesgue measurable real-valued function agrees almost everywhere with a Borel measurable function.
In spaces such as Lebesgue spaces, functions that agree almost everywhere are identified as a single equivalence class. Evaluation at an individual point is therefore not generally an operation on the equivalence class, even though measurable representatives exist.
Historical formulation
Émile Borel’s work on countably generated families of subsets of the real line established the class now called the Borel sets. Henri Lebesgue subsequently placed measurable functions at the center of his theory of integration by treating inverse images of intervals and approximation by simple functions as the relevant structural properties.
The later abstraction from subsets of Euclidean space to arbitrary sigma-algebras separated measurability from geometric regularity. In this formulation, a measurable function is a morphism of measurable spaces, while measures and integrals constitute additional structures placed on the domain. The abstract definition also supplied the common language used in probability theory, where a random variable is precisely a measurable function from a probability space to a measurable value space.