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Wikipedia / Measurable function

In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable.

Formal definition

are sets equipped with respective σ-algebras

is the σ-algebra generated by f

Term usage variations

or other topological spaces, the Borel algebra (generated by all the open sets) is a common choice. Some authors define measurable functions as exclusively real-valued ones with respect to the Borel algebra.

If the values of the function lie in an infinite-dimensional vector space, other non-equivalent definitions of measurability, such as weak measurability and Bochner measurability, exist.

Notable classes of measurable functions

  • Random variables are by definition measurable functions defined on probability spaces.

  • are Borel spaces 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 it is called a Borel section.

  • A Lebesgue measurable is the Borel algebra on the complex numbers Lebesgue measurable functions are of interest in mathematical analysis or the preimage of any open set being measurable. Continuous functions, monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functions of bounded variation are all Lebesgue measurable. is measurable if and only if the real and imaginary parts are measurable.

Properties of measurable functions

  • The sum and product of two complex-valued measurable functions are measurable. So is the quotient, so long as there is no division by zero.

  • Indeed, two Lebesgue-measurable functions may be constructed in such a way as to make their composition non-Lebesgue-measurable.

  • The (pointwise) supremum, infimum, limit superior, and limit inferior of a sequence (viz., countably many) of real-valued measurable functions are all measurable as well.

  • The pointwise is non-metrizable. The corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence.

Non-measurable functions

Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to prove the existence of non-measurable functions. Such proofs rely on the axiom of choice in an essential way, in the sense that Zermelo–Fraenkel set theory without the axiom of choice does not prove the existence of such functions.

In any measure space with a non-measurable set one can construct a non-measurable indicator function is equipped with the usual Borel algebra  

See also

  • - Vector spaces of measurable functions: the L^p spaces

Notes