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.

1 Special measurable functions
2 Properties of measurable functions
3 Non-measurable functions
4 See also
5 Notes

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 $Σ 1 / Σ 2$ and a function g is measurable $Σ 2 / Τ$ , then the composition $g \circ f$ is measurable $Σ 1 / T$ . ^[1]
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.

Non-measurable functions

Not all functions are measurable. For example, if $A$ is a non-measurable subset of the real line $\R$ , then its indicator function $1 A (x)$ is non-measurable.