outer measure

English

outer measure (plural outer measures)

(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 B_j's then this function applied to A is less-than-or-equal to the sum of the applications of this function to the B_j's).

Gerald B. Folland (1999) Real Analysis : Modern Techniques and Their Applications, Second edition, New York: John Wiley & Sons, Inc., →ISBN, OCLC 803988029, §1.4, pages 28, 29, 31