Measurable function

2020 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