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 ${\displaystyle (X,\Sigma )}$ 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. ${\displaystyle \forall E\in \mathrm {T} }$

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

That is, ${\displaystyle \sigma (f)\subseteq \Sigma }$, where ${\displaystyle \sigma (f)}$ is the σ-algebra generated by f. 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.

In any measure space ${\displaystyle (X,\Sigma )}$ with a non-measurable set ${\displaystyle A\subset X}$, ${\displaystyle A\notin \Sigma }$, one can construct a non-measurable indicator function:

${\displaystyle \mathbf {1} _{A}:(X,\Sigma )\to \mathbb {R} ,\quad \mathbf {1} _{A}(x)={\begin{cases}1&{\text{ if }}x\in A\\0&{\text{ otherwise}},\end{cases}}}$

where ${\displaystyle \mathbb {R} }$ is equipped with the usual Borel algebra. This is a non-measurable function since the preimage of the measurable set ${\displaystyle \{1\}}$ is the non-measurable ${\displaystyle A}$.  

As another example, any non-constant function ${\displaystyle f:X\to \mathbb {R} }$ is non-measurable with respect to the trivial ${\displaystyle \sigma }$-algebra ${\displaystyle \Sigma =\{\emptyset ,X\}}$, since the preimage of any point in the range is some proper, nonempty subset of ${\displaystyle X}$, which is not an element of the trivial ${\displaystyle \Sigma }$.

See also

• Vector spaces of measurable functions: the ${\displaystyle L^{p}}$ spaces
• Measure-preserving dynamical system

Notes

1. 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.

External links

• Measurable function at Encyclopedia of Mathematics
• Borel function at Encyclopedia of Mathematics