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. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any 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 for every $E\in \mathrm {T}$ the pre-image of $E$ under $f$ is in $\Sigma$ ; i.e. $\forall E\in \mathrm {T}$

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

That is, $\sigma (f)\subseteq \Sigma$ , where $\sigma (f)$ is the σ-algebra generated by f. 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 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 $(X,\Sigma )$ 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 $\alpha$ , 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 $f:X\to \mathbb {C}$ 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 $\Sigma _{3}\subseteq \Sigma _{2}$ . 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 $f_{n}:X\to Y$ 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

^ ^a ^b ^c ^d 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.

External links

[strichartz-1] Strichartz, Robert (2000). The Way of Analysis. Jones and Bartlett. ISBN 0-7637-1497-6.

[carothers-2] Carothers, N. L. (2000). Real Analysis. Cambridge University Press. ISBN 0-521-49756-6.

[folland-3] Folland, Gerald B. (1999). Real Analysis: Modern Techniques and their Applications. Wiley. ISBN 0-471-31716-0.

[royden-4] Royden, H. L. (1988). Real Analysis. Prentice Hall. ISBN 0-02-404151-3.

[dudley-5] Dudley, R. M. (2002). Real Analysis and Probability (2 ed.). Cambridge University Press. ISBN 0-521-00754-2.

[aliprantis-6] 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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