In mathematics, measurable functions are well-behaved functions between measurable spaces. Functions studied in analysis that are not measurable are generally considered pathological. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ... Partial plot of a function f. ... In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ...

If Σ is a σ-algebra over a set X and Τ is a σ-algebra over Y, then a function f : X → Y is (Σ-)measurable if the preimage of every set in Τ is in Σ. In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ...

By convention, if Y is some topological space, such as the space of real numbers or the complex numbers , then the Borel σ-algebra generated by the open sets on Y is used, unless otherwise specified. The measurable space (X,Σ) is also called a Borel space in this case. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space X is a σ-algebra of subsets of X associated to the topology of X. In the mathematics literature, there are at least two inequivalent definitions of this σ-algebra: The minimal σ-algebra containing the open sets. ... In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ...

Special measurable functions

If (X,Σ) and (Y,Τ) are Borel spaces, a measurable function f is also called a Borel function. Continuous functions are Borel but not all Borel functions are continuous. In mathematics, measurable functions are well-behaved functions between measurable spaces. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...

Random variables are by definition measurable functions defined on sample spaces. A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... In probability theory, the sample space, often denoted S, Ω or U (for universe), of an experiment or random trial is the set of all possible outcomes. ...

In the category of measurable spaces, the measurable functions are the morphisms. If X=Y and Σ=Τ, a measurable function f is called an endomorphism or a measure-preserving or stationary transformation of the measure space (X,Σ,μ) if and only if the measure μ is invariant under composition with f, i.e. In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ... In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of ergodic theory. ... In mathematics, a measure is a function that assigns a number, e. ... Measure can mean: To perform a measurement. ...

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A stationary transformation f is ergodic if every set in Σ, T invariant under f almost everywhere, with respect to μ has measure 0 or 1, i.e. In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ...

where AΔB denotes the symmetric difference .

An equivalent statement is that every set in Σ, T invariant under f, with respect to μ has measure 0 or 1, i.e.

A stochastic process is stationary if the domain of the sample functions is a time interval and all the time-shift transformations are stationary. If given a stationary or ergodic transformation f^t for all time t where , a stationary or stationary ergodic process X can be constructed by defining a measurable function X₀, composing it with f^t. i.e. In the mathematics of probability, a stochastic process is a random function. ... In statistics and signal processing, a time series is a sequence of data points, measured typically at successive times, spaced apart at uniform time intervals. ... Please wikify (format) this article as suggested in the Guide to layout and the Manual of Style. ...

and so X maps the sample space to a functional space with domain t. Ergodic processes in general need not be stationary although processes generated this way with an ergodic transformation must be stationary ergodic. An ergodic process that is not stationary can, for example, be generated by running an ergodic Markov chain with an initial distribution other than its stationary distribution.

Properties of measurable functions

• The sum and product of two real valued measurable functions is measurable.
• The composition of two measurable functions may not be a measurable function.
• Only measurable functions can be Lebesgue integrated.
• A useful characterisation of Lebesgue measurable functions is that f is measurable if and only if mid{-g,f,g} is integrable for all non-negative Lebesgue integrable functions g.