outer measure,
n. 1. also called Caratheodory outer measure. a set function constructed preliminary to, and sharing many of the properties of, a measure. Explicitly, an outer measure, μ*, on a set S is an isotone, countably additive, extended-real-valued set function defined for all subsets of the set and assigning value zero to the empty set; that is
μ*(∅) = 0,
μ*(E) ≤ μ*(F) if F contains E,
2a. Lebesgue outer measure. the particular outer measure of a set in Euclidean n-space computed by taking the infimum of the sum of the volumes (content) of any Lebesgue covering of the set by countable families of open finite order intervals (boxes):
μ*(E) = inf { Σ|In| : E lies in ∪In }. The restriction of this function to the Lebesgue measurable subsets defines Lebesgue measure in n-space.
b. Jordan outer measure. an analogous measure defined using only finite covers. Compare inner measure.
μ*(∅) = 0,
μ*(E) ≤ μ*(F) if F contains E,
2a. Lebesgue outer measure. the particular outer measure of a set in Euclidean n-space computed by taking the infimum of the sum of the volumes (content) of any Lebesgue covering of the set by countable families of open finite order intervals (boxes):
μ*(E) = inf { Σ|In| : E lies in ∪In }. The restriction of this function to the Lebesgue measurable subsets defines Lebesgue measure in n-space.
b. Jordan outer measure. an analogous measure defined using only finite covers. Compare inner measure.