Lebesguen.

/ləˈbɛɡ/

Etymology: < the name of Henri Lebesgue (1875–1941), French mathematician.

Mathematics.

Used attributively to designate various concepts introduced by Lebesgue or arising out of his work.

a. Lebesgue integral n. a definite integral of a bounded measurable function defined in a particular way, namely, that obtained by subdividing the range of the function, multiplying the lower (or upper) bound of each subdivision by the measure of its inverse image under the function, summing the products so obtained, and taking the limit of the sum as the width of the subdivisions tends to zero; an analogous integral of an unbounded function.

ΘΚΠ

the world > relative properties > number > calculus > [noun] > integral calculus > integration or integrability > integral

fluent1706

integral1728

gamma function1834

surface integral1867

Riemann integral1894

Cauchy's integral1898

Lebesgue integral1905

Stieltjes integral1914

convolution1934

1905 W. H. Young in Philos. Trans. (Royal Soc.) A. 204 244 Lebesgue gives two definitions of his generalised integral, which I shall, for convenience, allude to as the Lebesgue integral.

1912 Bull. Amer. Math. Soc. 18 237 Suppose the function f(p) to be defined and limited and to have a double Lebesgue integral on the surface of a unit sphere.

1957 Encycl. Brit. XIII. 855/1 Lebesgue formulated, in brilliant memoirs of 1901 and 1902, a new definition of the definite integral... This ‘Lebesgue integral’ is one of the great achievements of modern real analysis.

b. Lebesgue measure n. the measure (measure n. 8h) of a given set in n-dimensional space as defined by Lebesgue in 1902 (' Annali di Matematica 7 236), being (in the case of n = 1) the greatest lower bound of the sum of the lengths of a countable collection of intervals containing the set, taken over all possible such collections.

ΘΚΠ

the world > relative properties > number > mathematical number or quantity > numerical arrangement > [noun] > set > property or measure of

power1903

potency1904

Lebesgue measure1929

1929 Encycl. Brit. XVIII. 117/2 The Lebesgue measure of a set of points may not exist, but it does exist for all ordinary point sets.

1971 Sci. Amer. Aug. 95/2 Since each point x belonging to the set I is considered to be an interval of length 0, the Lebesgue measure m of each point x is 0.

1980 A. J. Jones Game Theory ii. 110 This function..is continuous everywhere, and differentiable except on an uncountable set of Lebesgue measure zero.

This entry has not yet been fully updated (first published 1997; most recently modified version published online September 2018).

单词	lebesgue
释义	Lebesguen. /ləˈbɛɡ/ Etymology: < the name of Henri Lebesgue (1875–1941), French mathematician. Mathematics. Used attributively to designate various concepts introduced by Lebesgue or arising out of his work. a. Lebesgue integral n. a definite integral of a bounded measurable function defined in a particular way, namely, that obtained by subdividing the range of the function, multiplying the lower (or upper) bound of each subdivision by the measure of its inverse image under the function, summing the products so obtained, and taking the limit of the sum as the width of the subdivisions tends to zero; an analogous integral of an unbounded function. ΘΚΠ the world > relative properties > number > calculus > [noun] > integral calculus > integration or integrability > integral fluent1706 integral1728 gamma function1834 surface integral1867 Riemann integral1894 Cauchy's integral1898 Lebesgue integral1905 Stieltjes integral1914 convolution1934 1905 W. H. Young in Philos. Trans. (Royal Soc.) A. 204 244 Lebesgue gives two definitions of his generalised integral, which I shall, for convenience, allude to as the Lebesgue integral. 1912 Bull. Amer. Math. Soc. 18 237 Suppose the function f(p) to be defined and limited and to have a double Lebesgue integral on the surface of a unit sphere. 1957 Encycl. Brit. XIII. 855/1 Lebesgue formulated, in brilliant memoirs of 1901 and 1902, a new definition of the definite integral... This ‘Lebesgue integral’ is one of the great achievements of modern real analysis. b. Lebesgue measure n. the measure (measure n. 8h) of a given set in n-dimensional space as defined by Lebesgue in 1902 (' Annali di Matematica 7 236), being (in the case of n = 1) the greatest lower bound of the sum of the lengths of a countable collection of intervals containing the set, taken over all possible such collections. ΘΚΠ the world > relative properties > number > mathematical number or quantity > numerical arrangement > [noun] > set > property or measure of power1903 potency1904 Lebesgue measure1929 1929 Encycl. Brit. XVIII. 117/2 The Lebesgue measure of a set of points may not exist, but it does exist for all ordinary point sets. 1971 Sci. Amer. Aug. 95/2 Since each point x belonging to the set I is considered to be an interval of length 0, the Lebesgue measure m of each point x is 0. 1980 A. J. Jones Game Theory ii. 110 This function..is continuous everywhere, and differentiable except on an uncountable set of Lebesgue measure zero. This entry has not yet been fully updated (first published 1997; most recently modified version published online September 2018). < n.1905
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