# Measurable function

In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable, analogous to the definition that a function between topological spaces is continuous if it preserves the topological structure: 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.

## Formal definition

Let ${\displaystyle (X,\Sigma )}$  and ${\displaystyle (Y,\mathrm {T} )}$  be measurable spaces, meaning that ${\displaystyle X}$  and ${\displaystyle Y}$  are sets equipped with respective ${\displaystyle \sigma }$ -algebras ${\displaystyle \Sigma }$  and ${\displaystyle \mathrm {T} }$ . A function ${\displaystyle f:X\to Y}$  is said to be measurable if for every ${\displaystyle E\in \mathrm {T} }$  the pre-image of ${\displaystyle E}$  under ${\displaystyle f}$  is in ${\displaystyle \Sigma }$ ; i.e.

${\displaystyle f^{-1}(E):=\{x\in X|\;f(x)\in E\}\in \Sigma ,\;\;\forall E\in \mathrm {T} .}$

If ${\displaystyle f:X\to Y}$  is a measurable function, we will write

${\displaystyle f\colon (X,\Sigma )\rightarrow (Y,\mathrm {T} ).}$

to emphasize the dependency on the ${\displaystyle \sigma }$ -algebras ${\displaystyle \Sigma }$  and ${\displaystyle \mathrm {T} }$ .

## Term usage variations

The choice of ${\displaystyle \sigma }$ -algebras in the definition above is sometimes implicit and left up to the context. For example, for ${\displaystyle {\mathbb {R} }}$ , ${\displaystyle {\mathbb {C} }}$ , or other topological spaces, 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.[1]

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 ${\displaystyle (X,\Sigma )}$  and ${\displaystyle (Y,T)}$  are Borel spaces, a measurable function ${\displaystyle f:(X,\Sigma )\to (Y,T)}$  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 ${\displaystyle Y{\xrightarrow {~\pi ~}}X}$ , it is called a Borel section.
• A Lebesgue measurable function is a measurable function ${\displaystyle f:(\mathbb {R} ,{\mathcal {L}})\to (\mathbb {C} ,{\mathcal {B}}_{\mathbb {C} })}$ , where ${\displaystyle {\mathcal {L}}}$  is the ${\displaystyle \sigma }$ -algebra of Lebesgue measurable sets, and ${\displaystyle {\mathcal {B}}_{\mathbb {C} }}$  is the Borel algebra on the complex numbers ${\displaystyle \mathbb {C} }$ . Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated. In the case ${\displaystyle f:X\to \mathbb {R} }$ , ${\displaystyle f}$  is Lebesgue measurable iff ${\displaystyle \{f>\alpha \}=\{x\in X:f(x)>\alpha \}}$  is measurable for all ${\displaystyle \alpha \in \mathbb {R} }$ . This is also equivalent to any of ${\displaystyle \{f\geq \alpha \},\{f<\alpha \},\{f\leq \alpha \}}$  being measurable for all ${\displaystyle \alpha }$ , 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.[2] A function ${\displaystyle f:X\to \mathbb {C} }$  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.[3] So is the quotient, so long as there is no division by zero.[1]
• If ${\displaystyle f:(X,\Sigma _{1})\to (Y,\Sigma _{2})}$  and ${\displaystyle g:(Y,\Sigma _{2})\to (Z,\Sigma _{3})}$  are measurable functions, then so is their composition ${\displaystyle g\circ f:(X,\Sigma _{1})\to (Z,\Sigma _{3})}$ .[1]
• If ${\displaystyle f:(X,\Sigma _{1})\to (Y,\Sigma _{2})}$  and ${\displaystyle g:(Y,\Sigma _{3})\to (Z,\Sigma _{4})}$  are measurable functions, their composition ${\displaystyle g\circ f:X\to Z}$  need not be ${\displaystyle (\Sigma _{1},\Sigma _{4})}$ -measurable unless ${\displaystyle \Sigma _{3}\subseteq \Sigma _{2}}$ . Indeed, two Lebesgue-measurable functions may be constructed in such a way as to make their composition non-Lebesgue-measurable.
• The (pointwise) supremum, infimum, limit superior, and limit inferior of a sequence (viz., countably many) of real-valued measurable functions are all measurable as well.[1][4]
• The pointwise limit of a sequence of measurable functions ${\displaystyle f_{n}:X\to Y}$  is measurable, where ${\displaystyle Y}$  is a metric space (endowed with the Borel algebra). This is not true in general if ${\displaystyle Y}$  is non-metrizable. Note that the corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence.[5][6]

## 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 ${\displaystyle (X,\Sigma )}$  is some measurable space and ${\displaystyle A\subset X}$  is a non-measurable set, i.e. if ${\displaystyle A\notin \Sigma }$ , then the indicator function ${\displaystyle \mathbf {1} _{A}:(X,\Sigma )\to \mathbb {R} }$  is non-measurable (where ${\displaystyle \mathbb {R} }$  is equipped with the Borel algebra as usual), since the preimage of the measurable set ${\displaystyle \{1\}}$  is the non-measurable set ${\displaystyle A}$ . Here ${\displaystyle \mathbf {1} _{A}}$  is given by
${\displaystyle \mathbf {1} _{A}(x)={\begin{cases}1&{\text{ if }}x\in A;\\0&{\text{ otherwise}}.\end{cases}}}$
• Any non-constant function can be made non-measurable by equipping the domain and range with appropriate ${\displaystyle \sigma }$ -algebras. If ${\displaystyle f:X\to \mathbb {R} }$  is an arbitrary non-constant, real-valued function, then ${\displaystyle f}$  is non-measurable if ${\displaystyle X}$  is equipped with the trivial ${\displaystyle \sigma }$ -algebra ${\displaystyle \Sigma =\{\emptyset ,X\}}$ , since the preimage of any point in the range is some proper, nonempty subset of ${\displaystyle X}$ , and therefore does not lie in ${\displaystyle \Sigma }$ .

## Notes

1. ^ a b c d Strichartz, Robert (2000). The Way of Analysis. Jones and Bartlett. ISBN 0-7637-1497-6.
2. ^ Carothers, N. L. (2000). Real Analysis. Cambridge University Press. ISBN 0-521-49756-6.
3. ^ Folland, Gerald B. (1999). Real Analysis: Modern Techniques and their Applications. Wiley. ISBN 0-471-31716-0.
4. ^ Royden, H. L. (1988). Real Analysis. Prentice Hall. ISBN 0-02-404151-3.
5. ^ Dudley, R. M. (2002). Real Analysis and Probability (2 ed.). Cambridge University Press. ISBN 0-521-00754-2.
6. ^ Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis, A Hitchhiker’s Guide (3 ed.). Springer. ISBN 978-3-540-29587-7.