English Dictionary

outer measure

名詞

(mathematical analysis, measure theory) A measure-like function derived from some premeasure as follows: when it is applied to a set E, it yields the infimum of the premeasures of all possible premeasurable covers of E. More abstractly, a measure-like function whose domain is the power set of some underlying set, which has the properties of (1) yielding zero when applied to the empty set, (2) being monotonic (if A is subset of B then this function applied to A is less-than-or-equal to this function applied to B), (3) being subadditive (if A is a countable sum of Bⱼ's then this function applied to A is less-than-or-equal to the sum of the applications of this function to the Bⱼ's).

日本語の意味

外側測度: 測度論において、あるプリアィメジャーから派生して定義される関数で、ある集合に対して、その集合を覆うすべてのプリアィメジャ可能な被覆のプリアィメジャーの下限を取るもの。 / 性質として、空集合に対してゼロを与え、単調性（ある集合が別の集合の部分集合なら、その外側測度もそれに対応して小さいか等しくなる）や、可算部分加法性（無限個の集合の和集合に対して、その外側測度は各集合の外側測度の和以上にならない）を満たす。

The outer measure assigns to each subset the infimum of the premeasures of all countable covers by premeasurable sets, ensuring monotonicity, subadditivity, and that the empty set has measure zero.

外測度は、全ての部分集合に定義された前測度から導かれる測度類似の関数であり、任意の集合に対してその集合を被覆する前測度可能な集合族の前測度の下限を与え、空集合に対して0を与え、単調性と可算部分加法性を満たす。

意味（1）

(mathematical analysis, measure theory) A measure-like function derived from some premeasure as follows: when it is applied to a set E, it yields the infimum of the premeasures of all possible premeasurable covers of E. More abstractly, a measure-like function whose domain is the power set of some underlying set, which has the properties of (1) yielding zero when applied to the empty set, (2) being monotonic (if A is subset of B then this function applied to A is less-than-or-equal to this function applied to B), (3) being subadditive (if A is a countable sum of Bⱼ's then this function applied to A is less-than-or-equal to the sum of the applications of this function to the Bⱼ's).

outer measures （ plural ）