Saturated measure
Encyclopedia
In mathematics
, a measure
is said to be saturated if every locally measurable set is also measurable. A set
, not necessarily measurable, is said to be locally measurable if for every measurable set 
of finite measure, 
is measurable. 
-finite measures, and measures arising as the restriction of outer measures
, are saturated.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
is said to be saturated if every locally measurable set is also measurable. A set
, not necessarily measurable, is said to be locally measurable if for every measurable set of finite measure, is measurable. -finite measures, and measures arising as the restriction of outer measuresOuter measure
In mathematics, in particular in measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. A general theory of outer measures was first introduced by...
, are saturated.