Negligible Set

In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose. As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integral of a measurable function.

Negligible sets define several useful concepts that can be applied in various situations, such as truth almost everywhere. In order for these to work, it is generally only necessary that the negligible sets form an ideal; that is, that the empty set be negligible, the union of two negligible sets be negligible, and any subset of a negligible set be negligible. For some purposes, we also need this ideal to be a sigma-ideal, so that countable unions of negligible sets are also negligible. If I and J are both ideals of subsets of the same set X, then one may speak of I-negligible and J-negligible subsets.

The opposite of a negligible set is a generic property, which has various forms.

Read more about Negligible SetExamples, Derived Concepts

Other articles related to "negligible set, set, sets, negligible, negligible sets":

Vitali Covering Lemma - Vitali Covering Theorem
... In the covering theorem, the aim is to cover, up to  a "negligible set", a given set E ⊆ Rd by a disjoint subcollection extracted from a Vitali covering for E a Vitali class or Vitali covering for E  is a ... In the classical setting of Vitali, the negligible set is a Lebesgue negligible set, but measures other than the Lebesgue measure, and spaces other than Rd have also ... is useful if is a Vitali covering for E  and if E  is contained in an open set Ω ⊆ Rd, then the subcollection of sets U  in that are contained in Ω  is also a ...
Negligible Set - Derived Concepts
... Let X be a set, and let I be an ideal of negligible subsets of X ... If p is a proposition about the elements of X, then p is true almost everywhere if the set of points where p is true is the complement of a negligible set ... paragraph precise, then, let X be N, and let the negligible sets be the finite sets ...

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