Let and be measurable spaces, meaning that and -algebras and. A function is said to be measurable if for every the pre-image of under is in ; i.e. That is,, where is the σ-algebra generated by f. If is a measurable function, we will write to emphasize the dependency on the -algebras and.
Term usage variations
The choice of -algebras in the definition above is sometimes implicit and left up to the context. For example, for,, or other topological spaces, the Borel algebra 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.
If and are Borel spaces, a measurable function 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.
A Lebesgue measurable function is a measurable function, where is the -algebra of Lebesgue measurable sets, and is the Borel algebra on the complex numbers. Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated. In the case, is Lebesgue measurable iff is measurable for all. This is also equivalent to any of being measurable for all, 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. A function is measurable iff 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.
If and are measurable functions, then so is their composition.
If and are measurable functions, their composition need not be -measurable unless. Indeed, two Lebesgue-measurable functions may be constructed in such a way as to make their composition non-Lebesgue-measurable.
Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to find non-measurable functions.
So long as there are non-measurable sets in a measure space, there are non-measurable functions from that space. If is some measurable space and is a non-measurable set, i.e. if, then the indicator function is non-measurable, since the preimage of the measurable set is the non-measurable set. Here is given by
Any non-constant function can be made non-measurable by equipping the domain and range with appropriate -algebras. If is an arbitrary non-constant, real-valued function, then is non-measurable if is equipped with the trivial -algebra, since the preimage of any point in the range is some proper, nonempty subset of, and therefore does not lie in.
