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, analogous to the definition that a function between topological spaces is continuous if it preserves the topological structure: the preimage of each 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.
Let and be measurable spaces, meaning that and are sets equipped with respective -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 (containing all the open sets) is a common choice. Some authors define measurable functions as exclusively real-valued ones with respect to the Borel algebra.
Notable classes of measurable functions
- Random variables are by definition measurable functions defined on probability spaces.
- 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.
- 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 limit of a sequence of measurable functions is measurable, where is a metric space (endowed with the Borel algebra). This is not true in general if is non-metrizable. Note that the corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence.
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 (where is equipped with the Borel algebra as usual), 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 .
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