Measurable function
2020 Mathematics Subject Classification: Primary: 28A20 [MSN][ZBL]
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
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for every real number , where
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This definition is equivalent to the following: A real-valued function is measurable if
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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.
References
[H] | P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802 |
[DS] | N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523 |
[KF] | A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) MR1025126 MR0708717 MR0630899 MR0435771 MR0377444 MR0234241 MR0215962 MR0118796 MR1530727 MR0118795 MR0085462 MR0070045 Zbl 0932.46001 Zbl 0672.46001 Zbl 0501.46001 Zbl 0501.46002 Zbl 0235.46001 Zbl 0103.08801 |
Measurable function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measurable_function&oldid=23633