Measurable function

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 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.

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 functions but not all Borel functions are continuous. However, a measurable function is nearly a continuous function; see Luzin's theorem.

If a Borel function happens to be a section of some map , it is called a Borel section.

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 Σ₁ / Σ₂ and a function g is measurable Σ₂ / Τ, then the composition is measurable Σ₁ / T. ^[1]
• The supremum of a countable family of measurable functions is measurable. If (f_n) is a sequence of measurable functions taking values in [−∞, +∞], then is again measurable.
• The pointwise limit of a sequence 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

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 , then its indicator function 1_A(x) is non-measurable.

See also

• Vector spaces of measurable functions: the L^p spaces
• Measure-preserving dynamical system

Notes

1. ^ Billingsley, Patrick (1995). Probability and Measure. Wiley. ISBN 0-471-00710-2.