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| {{Refimprove|date=December 2009}}
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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]].
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| Negligible sets define several useful concepts that can be applied in various situations, such as truth [[almost everywhere]].
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| 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.
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| For some purposes, we also need this ideal to be a [[sigma-ideal]], so that [[countable]] unions of negligible sets are also negligible.
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| 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.
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| The opposite of a negligible set is a [[generic property]], which has various forms.
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| == Examples ==
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| 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]].
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| Then the negligible sets form an ideal.
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| 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.
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| Or let ''X'' be an [[uncountable set]], and let a subset of ''X'' be negligible if it is [[countable set|countable]].
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| Then the negligible sets form a sigma-ideal.
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| 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]].
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| Then the negligible sets form a sigma-ideal.
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| 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.
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| 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 > 0, there exists a finite or countable collection ''I''<sub>1</sub>, ''I''<sub>2</sub>, … of (possibly overlapping) intervals satisfying:
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| : <math> A \subset \bigcup_{k} I_k </math> | |
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| and
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| : <math> \sum_{k} |I_k| < \epsilon .</math>
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| This is a special case of the preceding example, using [[Lebesgue measure]], but described in elementary terms.
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| 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]]).
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| Then the negligible sets form a sigma-ideal.
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| ''X'' is a ''[[Baire space]]'' if the [[interior (topology)|interior]] of every such negligible set is empty.
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| Let ''X'' be a [[directed set]], and let a subset of ''X'' be negligible if it has an [[upper bound]].
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| Then the negligible sets form an ideal.
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| The first example is a special case of this using the usual ordering of ''N''.
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| In a [[coarse structure]], the controlled sets are negligible.
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| ==Derived concepts== | |
| Let ''X'' be a [[set (mathematics)|set]], and let ''I'' be an ideal of negligible [[subset]]s of ''X''.
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| 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.
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| That is, ''p'' may not always be true, but it's false so rarely that this can be ignored for the purposes at hand.
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| 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.
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| To make the introductory paragraph precise, then, let ''X'' be '''N''', and let the negligible sets be the finite sets.
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| Then ''f'' and ''g'' are sequences.
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| If ''Y'' is a [[topological space]], then ''f'' and ''g'' have the same limit, or both have none.
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| (When you generalise this to a directed sets, you get the same result, but for [[net (mathematics)|net]]s.)
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| Or, let ''X'' be a measure space, and let negligible sets be the null sets.
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| If ''Y'' is the [[real line]] '''R''', then either ''f'' and ''g'' have the same integral, or neither integral is defined.
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| ==See also== | |
| * [[Negligible function]]
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| * [[Generic property]]
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| ==References== | |
| {{Reflist}}
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| {{DEFAULTSORT:Negligible Set}}
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| [[Category:Mathematical analysis]] | |
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