Lebesgue measure
in
A countably-additive measure which is an extension of the volume as a function of
-dimensional intervals to a wider class
of sets, namely the Lebesgue-measurable sets. The class
contains the class
of Borel sets (cf. Borel set) and consists of all sets of the form
where
,
and
. One has for any
,
![]() | (*) |
where the infimum is taken over all possible countable families of intervals such that
. Formula (*) makes sense for every
and defines a set function
(which coincides with
on
), called the outer Lebesgue measure. A set
belongs to
if and only if
![]() |
for every bounded interval ; for all
,
![]() |
and for all ,
![]() |
if , then the last equality is sufficient for the membership
; if
is an orthogonal operator in
and
, then
for any
. 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) |
[3] | P.R. Halmos, "Measure theory" , v. Nostrand (1950) |
[4] | A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) |
Comments
The Lebesgue measure is a very particular example of a Haar measure, of a product measure (when ) 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) |
Lebesgue measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_measure&oldid=15438