Measurable function

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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 R or the complex numbers 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

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.

<math>(\forall A\in\Sigma)(\mu(f(A))=\mu(A))</math>.

A stationary transformation f is ergodic if every set in Σ invariant under f almost everywhere with respect to μ has measure 0 or 1, i.e.

<math>(\forall A\in\Sigma)\left(\mu(f(A)\oplus A)=0 \implies \mu(A)\in\{0,1\}\right)</math>

where <math>A\oplus B</math> denotes the symmetric difference <math>(A\cup B) \backslash (A \cap B)</math>.

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 <math>f^t</math> for all time t where <math>(\forall t,t_1) (f^t=f^{t_1}\circ f^{t-t_1})</math>, a stationary or stationary ergodic process <math>X</math> can be constructed by defining a measurable function <math>X_0</math>, composing it with <math>f^t</math>. i.e.

<math>X(\omega):= X_0(f^t(\omega))\quad \forall \omega\in\Omega</math>

and so <math>X</math> 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 composition of two measurable functions is measurable
• Only measurable functions can be 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 integrable functions g.

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