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Concept# Negligible set

Summary

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.
Let X be the set N of natural numbers, and let a subset of N be negligible if it is finite.
Then the negligible sets form an ideal.
This idea can be applied to any infinite set; but if applied to a finite set, every subset will be negligible, which is not a very useful notion.
Or let X be an uncountable set, and let a subset of X be negligible if it is countable.
Then the negligible sets form a sigma-ideal.
Let X be a measurable space equipped with a measure m, and let a subset of X be negligible if it is m-null.
Then the negligible sets form a sigma-ideal.
Every sigma-ideal on X can be recovered in this way by placing a suitable measure on X, although the measure may be rather pathological.
Let X be the set R of real numbers, and let a subset A of R be negligible if for each ε > 0, there exists a finite or countable collection I1, I2, ... of (possibly overlapping) intervals satisfying:
and
This is a special case of the preceding example, using Lebesgue measure, but described in elementary terms.

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In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called nonmeagre, or of the second category. See below for definitions of other related terms. The meagre subsets of a fixed space form a σ-ideal of subsets; that is, any subset of a meagre set is meagre, and the union of countably many meagre sets is meagre.

In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if is a set, "almost all elements of " means "all elements of but those in a negligible subset of ". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite, countable, or null. In contrast, "almost no" means "a negligible quantity"; that is, "almost no elements of " means "a negligible quantity of elements of ". Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) except for finitely many".

In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of measure zero, and is analogous to the notion of almost surely in probability theory. More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero, or equivalently, if the set of elements for which the property holds is conull.

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