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.

Let and be measurable spaces, meaning that and are sets equipped with respective -algebras and A function is said to be measurable if for every the pre-image of under is in ; that is, for all

That is, where is the σ-algebra generated by f. If is a measurable function, we will write

to emphasize the dependency on the -algebras and

The choice of -algebras in the definition above is sometimes implicit and left up to the context. For example, for 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.

- Random variables are by definition measurable functions defined on probability spaces.
- If and 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 a map it is called a**Borel section**. - A Lebesgue measurable function is a measurable function where is the -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. In the case is Lebesgue measurable iff is measurable for all This is also equivalent to any of being measurable for all 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 is measurable iff the real and imaginary parts are measurable.

- 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 and are measurable functions, then so is their composition
^{ [1] } - If and are measurable functions, their composition need not be -measurable unless 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 is measurable, where is a metric space (endowed with the Borel algebra). This is not true in general if is non-metrizable. Note that the corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence.
^{ [5] }^{ [6] }

Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to prove the existence of non-measurable functions. Such proofs rely on the axiom of choice in an essential way, in the sense that Zermelo–Fraenkel set theory without the axiom of choice does not prove the existence of such functions.

In any measure space * with a non-measurable set one can construct a non-measurable indicator function: *

where is equipped with the usual Borel algebra. This is a non-measurable function since the preimage of the measurable set is the non-measurable

As another example, any non-constant function is non-measurable with respect to the trivial -algebra since the preimage of any point in the range is some proper, nonempty subset of which is not an element of the trivial

- Bochner measurable function
- Bochner space
- Lp space – Function spaces generalizing finite-dimensional p norm spaces - Vector spaces of measurable functions: the spaces
- Measure-preserving dynamical system – Subject of study in ergodic theory
- Vector measure
- Weakly measurable function

- 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.

In mathematics, specifically in measure theory, a **Borel measure** on a topological space is a measure that is defined on all open sets. Some authors require additional restrictions on the measure, as described below.

In mathematics, a **measure** on a set is a systematic way to assign a number to subsets of a set, intuitively interpreted as the size of the subset. Those sets which can be associated with such a number, we call **measurable sets**. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space. This assigns the usual length, area, or volume to certain subsets of the given Euclidean space. For instance, the Lebesgue measure of an interval of real numbers is its usual length, but also assigns numbers to other kinds of sets in a way that is consistent with the lengths of intervals.

In mathematical analysis, a **null set** is a set that has **measure zero**. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.

In probability and statistics, a **random variable**, **random quantity**, **aleatory variable**, or **stochastic variable** is described informally as a variable whose values depend on outcomes of a random phenomenon. The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a measurable function defined on a probability space that maps from the sample space to the real numbers.

In mathematical analysis and in probability theory, a **σ-algebra** on a set *X* is a collection Σ of subsets of *X* that includes *X* itself, is closed under complement, and is closed under countable unions.

In mathematics, a **Borel set** is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.

In mathematics, an infinite series of numbers is said to **converge absolutely** if the sum of the absolute values of the summands is finite. More precisely, a real or complex series is said to **converge absolutely** if for some real number . Similarly, an improper integral of a function, , is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if

In mathematical analysis, **Hölder's inequality**, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of *L ^{p}* spaces.

In mathematics, **Fatou's lemma** establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou.

In mathematical logic, **descriptive set theory** (**DST**) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic.

In probability theory, the **conditional expectation**, **conditional expected value**, or **conditional mean** of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of "conditions" is known to occur. If the random variable can take on only a finite number of values, the “conditions” are that the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of this probability space.

In mathematics, a **Radon measure**, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space *X* that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures.

In mathematics, a positive measure *μ* defined on a *σ*-algebra Σ of subsets of a set *X* is called a finite measure if *μ*(*X*) is a finite real number, and a set *A* in Σ is of finite measure if *μ*(*A*) < ∞*.* The measure *μ* is called **σ-finite** if *X* is the countable union of measurable sets with finite measure. A set in a measure space is said to have ** σ-finite measure** if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.e. all finite measures are σ-finite but there are (many) σ-finite measures that are not finite.

In probability theory, a **standard probability space**, also called **Lebesgue–Rokhlin probability space** or just **Lebesgue space** is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940. Informally, it is a probability space consisting of an interval and/or a finite or countable number of atoms.

In mathematics—specifically, in functional analysis—a **weakly measurable function** taking values in a Banach space is a function whose composition with any element of the dual space is a measurable function in the usual (strong) sense. For separable spaces, the notions of weak and strong measurability agree.

In probability theory, a **Markov kernel** is a map that in the general theory of Markov processes, plays the role that the transition matrix does in the theory of Markov processes with a finite state space.

In mathematics, a **real-valued function** is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.

In probability theory, the **Doob–Dynkin lemma**, named after Joseph L. Doob and Eugene Dynkin, characterizes the situation when one random variable is a function of another by the inclusion of the -algebras generated by the random variables. The usual statement of the lemma is formulated in terms of one random variable being measurable with respect to the -algebra generated by the other.

The **Engelbert–Schmidt zero–one law** is a theorem that gives a mathematical criterion for an event associated with a continuous, non-decreasing additive functional of Brownian motion to have probability either 0 or 1, without the possibility of an intermediate value. This zero-one law is used in the study of questions of finiteness and asymptotic behavior for stochastic differential equations. This 0-1 law, published in 1981, is named after Hans-Jürgen Engelbert and the probabilist Wolfgang Schmidt.

In measure theory, **projection maps ** often appear when working with product spaces: The product sigma-algebra of measurable spaces is defined to be the finest such that the projection mappings will be measurable. Sometimes for some reasons product spaces are equipped with sigma-algebra different than *the* product sigma-algebra. In these cases the projections need not be measurable at all.

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