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.
|||P.R. Halmos, "Measure theory" , v. Nostrand (1950)|
|||N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)|
|||A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)|
Measurable function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measurable_function&oldid=11460