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In [[mathematics]], a '''negligible set''' is a set that is small enough that it can be ignored for some purpose.
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As common examples, [[finite set]]s can be ignored when studying the [[limit of a sequence]], and [[null set]]s can be ignored when studying the [[integral (measure theory)|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 (set theory)|ideal]]; that is, that the [[empty set]] be negligible, the [[union (set theory)|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 [[subset]]s of the same [[set (mathematics)|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.
 
== Examples ==
 
Let ''X'' be the set '''N''' of [[natural number]]s, and let a subset of '''N''' be negligible [[iff|if]] it is [[finite set|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 set|countable]].
Then the negligible sets form a sigma-ideal.
 
Let ''X'' be a [[measurable space]] equipped with a [[measure (mathematics)|measure]] ''m,'' and let a subset of ''X'' be negligible if it is ''m''-[[null set|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 number]]s, and let a subset ''A'' of '''R''' be negligible<ref>Billingsley, P. Probability and Measure. Wiley-Interscience, Third Edition, p.8, 1995.</ref> if for each ε&nbsp;&gt; 0, there exists a finite or countable collection ''I''<sub>1</sub>, ''I''<sub>2</sub>, … of (possibly overlapping) intervals satisfying:
 
: <math> A \subset \bigcup_{k} I_k </math>
 
and
 
: <math> \sum_{k} |I_k| < \epsilon .</math>
 
This is a special case of the preceding example, using [[Lebesgue measure]], but described in elementary terms.
 
Let ''X'' be a [[topological space]], and let a subset be negligible if it is of [[first category]], that is, if it is a countable union of [[nowhere-dense set]]s (where a set is nowhere-dense if it is not [[dense set|dense]] in any [[open set]]).
Then the negligible sets form a sigma-ideal.
''X'' is a ''[[Baire space]]'' if the [[interior (topology)|interior]] of every such negligible set is empty.
 
Let ''X'' be a [[directed set]], and let a subset of ''X'' be negligible if it has an [[upper bound]].
Then the negligible sets form an ideal.
The first example is a special case of this using the usual ordering of ''N''.
 
In a [[coarse structure]], the controlled sets are negligible.
 
==Derived concepts==
Let ''X'' be a [[set (mathematics)|set]], and let ''I'' be an ideal of negligible [[subset]]s 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 (set theory)|complement]] of a negligible set.
That is, ''p'' may not always be true, but it's false so rarely that this can be ignored for the purposes at hand.
 
If ''f'' and ''g'' are functions from ''X'' to the same space ''Y'', then ''f'' and ''g'' are ''equivalent'' if they are equal almost everywhere.
To make the introductory paragraph precise, then, let ''X'' be '''N''', and let the negligible sets be the finite sets.
Then ''f'' and ''g'' are sequences.
If ''Y'' is a [[topological space]], then ''f'' and ''g'' have the same limit, or both have none.
(When you generalise this to a directed sets, you get the same result, but for [[net (mathematics)|net]]s.)
Or, let ''X'' be a measure space, and let negligible sets be the null sets.
If ''Y'' is the [[real line]] '''R''', then either ''f'' and ''g'' have the same integral, or neither integral is defined.
 
==See also==
* [[Negligible function]]
* [[Generic property]]
 
==References==
{{Reflist}}
 
{{DEFAULTSORT:Negligible Set}}
[[Category:Mathematical analysis]]

Revision as of 14:33, 26 February 2014

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