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].

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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.

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