Lebesgue measure

in $ \mathbf R ^ {n} $

A countably-additive measure $ \lambda $ which is an extension of the volume as a function of $ n $-dimensional intervals to a wider class $ {\mathcal A} $ of sets, namely the Lebesgue-measurable sets. The class $ {\mathcal A} $ contains the class $ {\mathcal B} $ of Borel sets (cf. Borel set) and consists of all sets of the form $ A \cup B $ where $ B \subset B _ {1} $, $ A , B _ {1} \in {\mathcal B} $ and $ \lambda ( B _ {1} ) = 0 $. One has for any $ A \in {\mathcal A} $,

$$ \tag{* } \lambda ( A) = \inf \sum _ { j } \lambda ( I _ {j} ) , $$

where the infimum is taken over all possible countable families of intervals $ \{ I _ {j} \} $ such that $ A \subset \cup I _ {j} $. Formula (*) makes sense for every $ A \subset \mathbf R ^ {n} $ and defines a set function $ \lambda ^ {*} $( which coincides with $ \lambda $ on $ {\mathcal A} $), called the outer Lebesgue measure. A set $ A $ belongs to $ {\mathcal A} $ if and only if

$$ \lambda ( I) = \lambda ^ {*} ( A \cap I ) + \lambda ^ {*} ( I \setminus A ) $$

for every bounded interval $ I $; for all $ A \subset \mathbf R ^ {n} $,

$$ \lambda ^ {*} ( A) = \inf \{ {\lambda ( U ) } : {A \subset U , U \textrm{ is open} } \} , $$

and for all $ A \in {\mathcal A} $,

$$ \lambda ( A) = \lambda ^ {*} ( A) = \ \sup \{ {\lambda ( F ) } : {A \supset F , F \textrm{ is compact} } \} ; $$

if $ \lambda ^ {*} ( A) < \infty $, then the last equality is sufficient for the membership $ A \in {\mathcal A} $; if $ O $ is an orthogonal operator in $ \mathbf R ^ {n} $ and $ a \in \mathbf R ^ {n} $, then $ \lambda ( OA + a ) = \lambda ( A) $ for any $ A \in {\mathcal A} $. The Lebesgue measure was introduced by H. Lebesgue [1].

References

[1]	H. Lebesgue, "Intégrale, longeur, aire" , Univ. Paris (1902) (Thesis)
[2]	S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) MR0167578 Zbl 1196.28001 Zbl 0017.30004 Zbl 63.0183.05
[3]	P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[4]	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

Comments

The Lebesgue measure is a very particular example of a Haar measure, of a product measure (when $ n > 1 $) and of a Hausdorff measure. Actually it is historically the first example of such measures.

References

[a1]	E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202