In mathematics, measurable functions are well-behaved functions between measurable spaces. Functions studied in analysis that are not measurable are generally considered pathological.

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.

By convention, if Y is some topological space, such as the space of real numbers $\backslash mathbb\{R\}$ or the complex numbers $\backslash mathbb\{C\}$, 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.

Special Measurable Functions

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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.

Random variables are by definition measurable functions defined on sample spaces.

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.

$(\backslash forall\; A\backslash in\backslash Sigma)(\backslash mu(f(A))=\backslash mu(A))$.

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.

$(\backslash forall\; A\backslash in\backslash Sigma)\backslash left(\backslash mu(f(A)\backslash Delta\; A)=0\; \backslash implies\; \backslash mu(A)\backslash in\backslash \{0,1\backslash \}\backslash right)$

where $A\backslash Delta\; B$ denotes the symmetric difference $(A\backslash cup\; B)\; \backslash backslash\; (A\; \backslash cap\; B)$.

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

$(\backslash forall\; A\backslash in\backslash Sigma)\; (\backslash mu(f(A))=0)\; \backslash implies\; \backslash mu(A)\backslash in\backslash \{0,1\backslash \}$

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 $(\backslash forall\; t,t\_1)\; (f^t=f^\{t\_1\}\backslash circ\; f^\{t-t\_1\})$, a stationary or stationary ergodic process $X$ can be constructed by defining a measurable function $X\_0$, composing it with $f^t$. i.e.

$X(\backslash omega):=\; X\_0(f^t(\backslash omega))\backslash quad\; \backslash forall\; \backslash omega\backslash in\backslash Omega$

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

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• The composition of two measurable functions is measurable
• 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.

Measure theory

messbare Funktion | Funzione misurabile | Función medible | funkcja mierzalna | Измеримая функция