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 measurable Σ/Τ 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 $mathbb\{R\}$ or the complex numbers $mathbb\{C\}$, 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.

If it is clear from the context what Τ and/or Σ are, then the function f may be (and usually is) called Σ-measurable or simply measurable.

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. However, a measurable function is nearly a continuous function; see Luzin's theorem.

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

Properties of measurable functions

• The sum and product of two real-valued measurable functions are measurable.
• If a function f is measurable $Sigma\_1/Sigma\_2$ and a function g is measurable $Sigma\_2/Tau$, then the composition $g\; circ\; f$ is measurable $Sigma\_1/T$.
• The pointwise limit of measurable functions is measurable. (The corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence.)
• Only measurable functions can be Lebesgue integrated.
• A Lebesgue-measurable function is a real function f : R → R such that for every real number a, the set

$\{x\; in\; R\; :\; f(x)>a\; \}$

is a Lebesgue-measurable set. 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.

Stationary transformations

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, that is, for all A in Σ one has

$mu(f(A))=mu(A).$

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.

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

where $ADelta\; B$ denotes the symmetric difference $(Acup\; B)\; backslash\; (A\; cap\; B)$.

An equivalent statement is that every set in Σ, T invariant under f, with respect to μ has measure 0 or 1, that is to say, for all A in Σ one has

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

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

$X(omega):=\; X\_0(f^t(omega))quad\; forall\; omegainOmega$

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.

Notes

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Last updated on Monday January 28, 2008 at 13:02:09 PST (GMT -0800)
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