Originally, a measurable function was understood to be a function of a real variable with the property that for every the set of points at which is a (Lebesgue-) measurable set. A measurable function on an interval can be made continuous on by changing its values on a set of arbitrarily small measure; this is the so-called -property of measurable functions (N.N. Luzin, 1913, cf. also Luzin -property).
A measurable function on a space is defined relative to a chosen system of measurable sets in . If is a -ring, then a real-valued function on is said to be a measurable function if
for every real number , where
This definition is equivalent to the following: A real-valued function is measurable if
for every Borel set . When is a -algebra, a function is measurable if (or ) is measurable. The class of measurable functions is closed under the arithmetical and lattice operations; that is, if , are measurable, then , , , and ( real) are measurable; and are also measurable. A complex-valued function is measurable if its real and imaginary parts are measurable. A generalization of the concept of a measurable function is that of a measurable mapping from one measurable space to another.
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Measurable function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Measurable_function&oldid=26623