In mathematics and in particular measure theory, a measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable, analogously to the definition that a function between topological spaces is continuous if 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.
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✪ Measure Theory for Applied Research (Class.4: Measurable Functions)

✪ Measurable Functions

✪ Mod05 Lec14 Measurable functions

✪ Lebesgue measurable functions and basic properties (MAT)

✪ Lecture 10: Integration: measurable and simple functions
Transcription
Contents
Formal definition
Let and be measurable spaces, meaning that and are sets equipped with respective algebras and . A function is said to be measurable if the preimage of under is in for every ; i.e.
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 space, the Borel algebra (containing all the open sets) is a common choice. Some authors define measurable functions as exclusively realvalued ones with respect to the Borel algebra.^{[1]}
If the values of the function lie in an infinitedimensional vector space, other nonequivalent 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.
 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, Riemannintegrable functions, and functions of bounded variation are all Lebesgue measurable.^{[2]} A function is measurable iff the real and imaginary parts are measurable.
Properties of measurable functions
 The sum and product of two complexvalued measurable functions are measurable.^{[3]} So is the quotient, so long as there is no division by zero.^{[1]}
 If and are measurable functions, then so is their composition .^{[1]}
 If and are measurable functions, their composition need not be measurable unless . Indeed, two Lebesguemeasurable functions may be constructed in such a way as to make their composition nonLebesguemeasurable.
 The (pointwise) supremum, infimum, limit superior, and limit inferior of a sequence (viz., countably many) of realvalued measurable functions are all measurable as well.^{[1]}^{[4]}
 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 nonmetrizable. Note that the corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence.^{[5]}^{[6]}
Nonmeasurable functions
Realvalued functions encountered in applications tend to be measurable; however, it is not difficult to find nonmeasurable functions.
 So long as there are nonmeasurable sets in a measure space, there are nonmeasurable functions from that space. If is some measurable space and is a nonmeasurable set, i.e. if , then the indicator function is nonmeasurable (where is equipped with the Borel algebra as usual), since the preimage of the measurable set is the nonmeasurable set . Here is given by
 Any nonconstant function can be made nonmeasurable by equipping the domain and range with appropriate algebras. If is an arbitrary nonconstant, realvalued function, then is nonmeasurable 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 .
See also
 Vector spaces of measurable functions: the spaces
 Measurepreserving dynamical system
Notes
 ^ ^{a} ^{b} ^{c} ^{d} Strichartz, Robert (2000). The Way of Analysis. Jones and Bartlett. ISBN 0763714976.
 ^ Carothers, N. L. (2000). Real Analysis. Cambridge University Press. ISBN 0521497566.
 ^ Folland, Gerald B. (1999). Real Analysis: Modern Techniques and their Applications. Wiley. ISBN 0471317160.
 ^ Royden, H. L. (1988). Real Analysis. Prentice Hall. ISBN 0024041513.
 ^ Dudley, R. M. (2002). Real Analysis and Probability (2 ed.). Cambridge University Press. ISBN 0521007542.
 ^ Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis, A Hitchhiker’s Guide (3 ed.). Springer. ISBN 9783540295877.