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have the same limit, or both have none. (When you generalise this to a directed sets, you get the same result, but for
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Negligible sets define several useful concepts that can be applied in various situations, such as truth
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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 they are equal almost everywhere. To make the introductory paragraph precise, then, let
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be negligible if for each ε > 0, there exists a finite or countable collection
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of a negligible set be negligible. For some purposes, we also need this ideal to be a
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that is small enough that it can be ignored for some purpose. As common examples,
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305:. Then the negligible sets form a sigma-ideal. Every sigma-ideal on
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be a measure space, and let negligible sets be the null sets. If
708:(Third ed.). New York: John Wiley & Sons. p. 8.
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can be recovered in this way by placing a suitable measure on
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have the same integral, or neither integral is defined.
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This is a special case of the preceding example, using
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184:unions of negligible sets are also negligible. If
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451:{\displaystyle \sum _{k}|I_{k}|<\epsilon .}
172:of two negligible sets be negligible, and any
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484:(where a set is nowhere-dense if it is not
389:{\displaystyle A\subset \bigcup _{k}I_{k}}
16:Mathematical set regarded as insignificant
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109:Learn how and when to remove this message
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554:is a proposition about the elements of
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47:adding citations to reliable sources
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145:can be ignored when studying the
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34:needs additional citations for
576:of a negligible set. That is,
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680:Negligible function
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139:limit of a sequence
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99:December 2009
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32:This article
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41:Please help
36:verification
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647:.) Or, let
509:upper bound
320:be the set
236:be the set
215:-negligible
208:-negligible
178:sigma-ideal
135:finite sets
123:mathematics
741:Set theory
730:Categories
691:References
609:equivalent
574:complement
538:, and let
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69:newspapers
657:real line
443:ϵ
411:∑
368:⋃
364:⊂
273:countable
217:subsets.
182:countable
166:empty set
143:null sets
674:See also
562:is true
490:open set
228:Examples
147:integral
655:is the
635:, then
599:, then
558:, then
544:subsets
488:in any
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259:Or let
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83:scholar
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250:finite
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631:is a
550:. If
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518:In a
499:be a
486:dense
472:be a
282:be a
170:union
162:ideal
149:of a
129:is a
90:JSTOR
76:books
710:ISBN
666:and
645:nets
639:and
623:and
607:are
603:and
587:and
530:Let
495:Let
468:Let
440:<
399:and
316:Let
303:null
278:Let
232:Let
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62:news
615:be
583:If
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332:of
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131:set
121:In
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