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| In [[mathematics]], more precisely in [[measure theory]], an '''atom''' is a measurable set which has positive measure and contains no set of smaller but positive measure. A measure which has no atoms is called '''non-atomic''' or '''atomless'''.
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| ==Definition==
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| Given a [[measurable space]] <math>(X, \Sigma)</math> and a [[measure (mathematics)|measure]] <math>\mu</math> on that space, a set <math>A</math> in <math>\Sigma</math> is called an '''atom''' if
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| : <math> \mu (A) >0\, </math>
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| and for any measurable subset <math>B</math> of <math>A</math> with
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| : <math> \mu(A) > \mu (B) \, </math>
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| one has <math> \mu(B)=0.</math>
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| == Examples==
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| * Consider the set ''X''={1, 2, ..., 9, 10} and let the sigma-algebra <math>\Sigma</math> be the [[power set]] of ''X''. Define the measure <math>\mu</math> of a set to be its [[cardinality]], that is, the number of elements in the set. Then, each of the [[singleton (mathematics)|singleton]]s {''i''}, for ''i''=1,2, ..., 9, 10 is an atom.
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| * Consider the [[Lebesgue measure]] on the [[real line]]. This measure has no atoms.
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| == Non-atomic measures==
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| A measure which has no atoms is called '''non-atomic'''. In other words, a measure is non-atomic if for any measurable set <math>A</math> with <math> \mu (A) >0</math> there exists a measurable subset ''B'' of ''A'' such that
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| : <math> \mu(A) > \mu (B) > 0. \, </math>
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| A non-atomic measure with at least one positive value has an infinite number of distinct values, as starting with a set ''A'' with <math> \mu (A) >0</math> one can construct a decreasing sequence of measurable sets
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| :<math>A=A_1\supset A_2 \supset A_3 \supset \cdots</math>
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| such that
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| :<math>\mu(A)=\mu(A_1) > \mu(A_2) > \mu(A_3) > \cdots > 0. </math>
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| This may not be true for measures having atoms; see the first example above.
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| It turns out that non-atomic measures actually have a [[Continuum (theory)|continuum]] of values. It can be proved that if μ is a non-atomic measure and ''A'' is a measurable set with <math>\mu (A) >0,</math> then for any real number ''b'' satisfying
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| : <math>\mu (A) \geq b \geq0\, </math>
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| there exists a measurable subset ''B'' of ''A'' such that
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| : <math>\mu(B)=b.\,</math>
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| This theorem is due to [[Wacław Sierpiński]].<ref>{{cite journal |first=W. |last=Sierpinski |url=http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3125.pdf |title=Sur les fonctions d'ensemble additives et continues |journal=Fundamenta Mathematicae |volume=3 |issue= |pages=240–246 |year=1922|language=French }}</ref><ref>{{Cite book |last=Fryszkowski |first=Andrzej |title=Fixed Point Theory for Decomposable Sets (Topological Fixed Point Theory and Its Applications) |year=2005 |publisher=Springer |location=New York |isbn=1-4020-2498-3 |page=39}}</ref>
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| It is reminiscent of the [[intermediate value theorem]] for continuous functions.
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| '''Sketch of proof''' of Sierpiński's theorem on non-atomic measures. A slightly stronger statement, which however makes the proof easier, is that if <math>(X,\Sigma, \mu)</math> is a non-atomic measure space and <math>\mu(X)=c</math>, there exists a function <math>S:[0, c]\to\Sigma</math> that is monotone with respect to inclusion, and a right-inverse to <math>\mu:\Sigma\to[0,\,c]</math>. That is, there exists a one-parameter family of measurable sets S(t) such that for all <math>0\leq t \leq t'\leq c</math>
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| :<math>S(t)\subset S(t'),</math>
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| :<math>\mu\left (S(t)\right)=t.</math>
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| The proof easily follows from [[Zorn's lemma]] applied to the set of all monotone partial sections to <math>\mu</math> :
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| :<math>\Gamma:=\{S:D\to\Sigma\; :\; D\subset[0,\,c],\, S\; \mathrm{ monotone }, \forall t\in D\; (\mu\left (S(t)\right)=t)\},</math>
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| ordered by inclusion of graphs, <math>\mathrm{graph}(S)\subset \mathrm{graph}(S').</math> It's then standard to show that every chain in <math>\Gamma</math> has an upper bound in <math>\Gamma</math>, and that any maximal element of <math>\Gamma</math> has domain <math>[0,c],</math> proving the claim.
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| == See also ==
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| * [[Atom (order theory)]] — an analogous concept in order theory
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| * [[Dirac delta function]]
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| * [[Elementary event]], also known as an '''atomic event'''
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| == Notes==
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| <references />
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| ==References==
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| * {{Cite book | author=Bruckner, Andrew M.; Bruckner, Judith B.; Thomson, Brian S. | authorlink= | coauthors= | title=Real analysis | year=1997 | publisher=Prentice-Hall | location=Upper Saddle River, N.J. | isbn=0-13-458886-X | page=108}}
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| * {{Cite book | author=Butnariu, Dan; Klement, E. P. | authorlink= | coauthors= | title=Triangular norm-based measures and games with fuzzy coalitions | year=1993 | publisher=Kluwer Academic | location=Dordrecht | isbn=0-7923-2369-6 | page=87}}
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| [[Category:Measure theory]]
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Oscar is what my spouse enjoys to call me and I totally dig that title. My day job is a meter reader. South Dakota is where I've usually been living. Body developing is 1 of the things I love most.
My website: std home test