| 释义 |
Lebesgue, n. Math.|ləˈbɛg|
[The name of Henri Lebesgue (1875–1941), French mathematician.]
Used attrib. to designate various concepts introduced by Lebesgue or arising out of his work, as Lebesgue integral, 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; Lebesgue measure, the measure (*measure n. 2 g) of a given set in n-dimensional space as defined by Lebesgue in 1902 (Annali di Matematica VII. 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.
1905W. H. Young in Phil. Trans. R. Soc. A. CCIV. 244 Lebesgue gives two definitions of his generalised integral, which I shall, for convenience, allude to as the Lebesgue integral. 1912Bull. Amer. Math. Soc. XVIII. 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. 1929Encycl. Brit. XVIII. 117/2 The Lebesgue measure of a set of points may not exist, but it does exist for all ordinary point sets. 1957Ibid. 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. 1971Sci. 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. 1980A. J. Jones Game Theory ii. 110 This function..is continuous everywhere, and differentiable except on an uncountable set of Lebesgue measure zero. |