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.

Formal definition

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

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

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

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

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

Term usage variations

The choice of $\sigma$ -algebras in the definition above is sometimes implicit and left up to the context. For example, for ${\mathbb R}$ , ${{\mathbb C}}$ , 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.^[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 $(X,\Sigma )$ and $(Y,T)$ are Borel spaces, a measurable function $f:(X,\Sigma )\to (Y,T)$ 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 $Y{\xrightarrow {~\pi ~}}X$ , it is called a Borel section.
A Lebesgue measurable function is a measurable function $f:(\mathbb {R} ,{\mathcal {L}})\to (\mathbb {C} ,{\mathcal {B}}_{\mathbb {C} })$ , where ${\mathcal {L}}$ is the $\sigma$ -algebra of Lebesgue measurable sets, and ${\mathcal {B}}_{\mathbb {C} }$ is the Borel algebra on the complex numbers $\mathbb {C}$ . Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated. In the case $f:X\to\mathbb{R}$ , $f$ is Lebesgue measurable iff $\{f>\alpha \}=\{x\in X:f(x)>\alpha \}$ is measurable for all $\alpha \in \mathbb {R}$ . This is also equivalent to any of $\{f\geq \alpha \},\{f<\alpha \},\{f\leq \alpha \}$ being measurable for all $\alpha$ . Continuous functions, monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functions of bounded variation are all Lebesgue measurable.^[2] A function $f:X\to {\mathbb {C}}$ 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 $f:(X,\Sigma _{1})\to (Y,\Sigma _{2})$ and $g:(Y,\Sigma _{2})\to (Z,\Sigma _{3})$ are measurable functions, then so is their composition $g\circ f:(X,\Sigma _{1})\to (Z,\Sigma _{3})$ .^[1]
If $f:(X,\Sigma _{1})\to (Y,\Sigma _{2})$ and $g:(Y,\Sigma _{3})\to (Z,\Sigma _{4})$ are measurable functions, their composition $g\circ f:X\to Z$ need not be $(\Sigma _{1},\Sigma _{4})$ -measurable unless $\Sigma _{2}$ and $\Sigma _{3}$ 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.^[1]^[4]
The pointwise limit of a sequence of measurable functions $f_{n}:X\to Y$ is measurable, where $Y$ is a metric space (endowed with the Borel algebra). This is not true in general if $Y$ 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 $(X,\Sigma )$ is some measurable space and $A\subset X$ is a non-measurable set, i.e. if $A\notin \Sigma$ , then the indicator function $\mathbf {1} _{A}:(X,\Sigma )\to \mathbb {R}$ is non-measurable (where $\mathbb {R}$ is equipped with the Borel algebra as usual), since the preimage of the measurable set $\{1\}$ is the non-measurable set $A$ . Here $\mathbf {1} _{A}$ is given by

\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 $\sigma$ -algebras. If $f:X\to\mathbb{R}$ is an arbitrary non-constant, real-valued function, then $f$ is non-measurable if $X$ is equipped with the trivial $\sigma$ -algebra $\Sigma =\{\emptyset ,X\}$ , since the preimage of any point in the range is some proper, nonempty subset of $X$ , and therefore does not lie in $\Sigma$ .

Notes

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

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