Namespaces
Variants
Actions

Measurable function

From Encyclopedia of Mathematics
Revision as of 08:00, 6 June 2020 by Ulf Rehmann (talk | contribs) (tex encoded by computer)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search


2010 Mathematics Subject Classification: Primary: 28A20 [MSN][ZBL]

Originally, a measurable function was understood to be a function $ f ( x) $ of a real variable $ x $ with the property that for every $ a $ the set $ E _ {a} $ of points $ x $ at which $ f ( x) < a $ is a (Lebesgue-) measurable set. A measurable function on an interval $ [ x _ {1} , x _ {2} ] $ can be made continuous on $ [ x _ {1} , x _ {2} ] $ by changing its values on a set of arbitrarily small measure; this is the so-called $ C $- property of measurable functions (N.N. Luzin, 1913, cf. also Luzin $ C $- property).

A measurable function on a space $ X $ is defined relative to a chosen system $ A $ of measurable sets in $ X $. If $ A $ is a $ \sigma $- ring, then a real-valued function $ f $ on $ X $ is said to be a measurable function if

$$ R _ {f} \cap E _ {a} \in A $$

for every real number $ a $, where

$$ E _ {a} = \{ {x \in X } : {f ( x) < a } \} , $$

$$ R _ {f} = \{ x \in X: f ( x) \neq 0 \} . $$

This definition is equivalent to the following: A real-valued function $ f $ is measurable if

$$ R _ {f} \cap \{ {x \in X } : {f ( x) \in B } \} \in A $$

for every Borel set $ B $. When $ A $ is a $ \sigma $- algebra, a function $ f $ is measurable if $ E _ {a} $( or $ \{ {x \in X } : {f ( x) \in B } \} $) is measurable. The class of measurable functions is closed under the arithmetical and lattice operations; that is, if $ f _ {n} $, $ n = 1, 2 \dots $ are measurable, then $ f _ {1} + f _ {2} $, $ f _ {1} f _ {2} $, $ \max ( f _ {1} , f _ {2} ) $, $ \min ( f _ {1} , f _ {2} ) $ and $ af $( $ a $ real) are measurable; $ \overline{\lim\limits}\; f _ {n} $ and $ fnnme \underline{lim} f _ {n} $ 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
How to Cite This Entry:
Measurable function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measurable_function&oldid=47814
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article