Measurable function

In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the 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 continuous functions between topological spaces.

This definition can be deceptively simple, however, as special care must be taken regarding the σ-algebras involved. In particular, when a function is said to be Lebesgue measurable what is actually meant is that is a measurable function -- that is, the domain and range represent different σ-algebras on the same underlying set (here is the sigma algebra of Lebesgue measurable sets, and is the Borel algebra on ). As a result, the composition of Lebesgue-measurable functions need not be Lebesgue-measurable.

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 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 function to refer only to real-valued measurable functions with respect to the Borel algebra.^[1]

Non-measurable functions are generally considered pathological, at least in the field of analysis.

Special measurable functions

• If (X,Σ) and (Y,Τ) are Borel spaces, a measurable function 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 , it is called a Borel section.

• A Lebesgue measurable function is a measurable function , where is the sigma algebra of Lebesgue measurable sets, and is the Borel algebra on the complex numbers . Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated.

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

Properties of measurable functions

• The sum and product of two complex-valued measurable functions are measurable.^[2]. So is the quotient, so long as there is no division by zero.^[1]

• If a function f is measurable Σ₁ / Σ₂ and a function g is measurable Σ₂ / Τ, then the composition is measurable Σ₁ / T. ^[1][3] (Here "measurable Σ₁ / Σ₂" means that is measurable.) In other words, the composition of measurable functions is measurable, so long as the appropriate σ-algebras match up (see the caveat regarding Lebesgue-measurable functions in the introduction.)

• The (pointwise) supremum, infimum, limit superior, and limit inferior of a sequence of real-valued measurable functions are all measurable as well.^[1][4]

• The pointwise limit of a sequence of measurable functions is measurable. (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) is some measurable space and is a non-measurable set, i.e. if , then the indicator function is non-measurable (where is equipped with the Borel algebra as usual), since the preimage of the measurable set {1} is the non-measurable set A. Here 1_A is given by

• Any non-constant function can be made non-measurable by equipping the domain and range with appropriate σ-algebras. If 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.

See also

• Vector spaces of measurable functions: the L^p spaces
• Measure-preserving dynamical system

Notes

1. ^ ^{a b c d} Strichartz, Robert (2000). The Way of Analysis. Jones and Bartlett. ISBN 0-7637-1497-6. 
2. ^ Folland, Gerald B. (1999). Real Analysis: Modern Techniques and their Applications. Wiley. ISBN 0471317160. 
3. ^ Billingsley, Patrick (1995). Probability and Measure. Wiley. ISBN 0-471-00710-2. 
4. ^ Royden, H. L. (1988). Real Analysis. Prentice Hall. ISBN 0-02-404151-3.