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[edit]

[1] |  H. Lebesgue, "Intégrale, longeur, aire" , Univ. Paris (1902) (Thesis)

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[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[edit]

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[edit]

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

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