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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 positive measure. A measure which has no atoms is called non-atomic or atomless.

Definition

Given a measurable space $(X,\Sigma )$ and a measure $\mu$ on that space, a set $A\subset X$ in $\Sigma$ is called an atom if

\mu (A)>0

and for any measurable subset $B\subset A$ with

\mu (B)<\mu (A)

the set $B$ has measure zero.

If $A$ is an atom, all the subsets in the $\mu$ -equivalence class $[A]$ of $A$ are atoms, and $[A]$ is called an atomic class. If $\mu$ is a $\sigma$ -finite measure, there are countable many atomic classes.

Examples

Consider the set X={1, 2, ..., 9, 10} and let the sigma-algebra $\Sigma$ be the power set of X. Define the measure $\mu$ of a set to be its cardinality, that is, the number of elements in the set. Then, each of the singletons {i}, for i=1,2, ..., 9, 10 is an atom.
Consider the Lebesgue measure on the real line. This measure has no atoms.

Atomic measures

A measure $\mu$ on a measurable space $(X,\Sigma )$ is called atomic or purely atomic if every measurable set of positive measure contains an atom.

Discrete measures

A $\sigma$ -finite measure $\mu$ is called discrete if it is atomic and the intersection of the atoms of any atomic class is non empty. It is equivalent to say that $\mu$ is the weighted sum of countably many Dirac measures, that is, there is a sequence $x_{1},x_{2},...$ of points in $X$ , and a sequence $c_{1},c_{2},...$ of positive real numbers (the weights) such that $\mu =\sum _{k=1}^{\infty }c_{k}\delta _{x_{k}}$ , which means that $\mu (A)=\sum _{k=1}^{\infty }c_{k}\delta _{x_{k}}(A)$ for every $A\in \Sigma$ . We can chose each point $x_{n}$ to be a common point of the atoms in the $n$ -th atomic class.

A discrete measure is atomic but the inverse implication fails: take $X=[0,1]$ , $\Sigma$ the $\sigma$ -algebra of countable and co-countable subsets, $\mu =0$ in countable subsets and $\mu =1$ in co-countable subsets. Then there is a single atomic class, the one formed by the co-countable subsets. The measure $\mu$ is atomic but the intersection of the atoms in the unique atomic class is empty and $\mu$ can't be put as a sum of Dirac measures.

Non-atomic measures

A measure which has no atoms is called non-atomic or diffuse. In other words, a measure $\mu$ is non-atomic if for any measurable set $A$ with $\mu (A)>0$ there exists a measurable subset B of A such that

\mu (A)>\mu (B)>0.\,

A non-atomic measure with at least one positive value has an infinite number of distinct values, as starting with a set A with $\mu (A)>0$ one can construct a decreasing sequence of measurable sets

A=A_{1}\supset A_{2}\supset A_{3}\supset \cdots

such that

\mu (A)=\mu (A_{1})>\mu (A_{2})>\mu (A_{3})>\cdots >0.

This may not be true for measures having atoms; see the first example above.

It turns out that non-atomic measures actually have a continuum of values. It can be proved that if μ is a non-atomic measure and A is a measurable set with $\mu (A)>0,$ then for any real number b satisfying

\mu (A)\geq b\geq 0\,

there exists a measurable subset B of A such that

\mu (B)=b.\,

This theorem is due to Wacław Sierpiński.^[1]^[2] It is reminiscent of the intermediate value theorem for continuous functions.

Sketch of proof of Sierpiński's theorem on non-atomic measures. A slightly stronger statement, which however makes the proof easier, is that if $(X,\Sigma ,\mu )$ is a non-atomic measure space and $\mu (X)=c$ , there exists a function $S:[0,c]\to \Sigma$ that is monotone with respect to inclusion, and a right-inverse to $\mu :\Sigma \to [0,\,c]$ . That is, there exists a one-parameter family of measurable sets S(t) such that for all $0\leq t\leq t'\leq c$

S(t)\subset S(t'),

\mu \left(S(t)\right)=t.

The proof easily follows from Zorn's lemma applied to the set of all monotone partial sections to $\mu$ :

\Gamma :=\{S:D\to \Sigma \;:\;D\subset [0,\,c],\,S\;\mathrm {monotone} ,\forall t\in D\;(\mu \left(S(t)\right)=t)\},

ordered by inclusion of graphs, $\mathrm {graph} (S)\subset \mathrm {graph} (S').$ It's then standard to show that every chain in $\Gamma$ has an upper bound in $\Gamma$ , and that any maximal element of $\Gamma$ has domain $[0,c],$ proving the claim.

Notes

^ Sierpinski, W. (1922). "Sur les fonctions d'ensemble additives et continues" (PDF). Fundamenta Mathematicae (in French). 3: 240–246.
^ Fryszkowski, Andrzej (2005). Fixed Point Theory for Decomposable Sets (Topological Fixed Point Theory and Its Applications). New York: Springer. p. 39. ISBN 1-4020-2498-3.

References

Bruckner, Andrew M.; Bruckner, Judith B.; Thomson, Brian S. (1997). Real analysis. Upper Saddle River, N.J.: Prentice-Hall. p. 108. ISBN 0-13-458886-X.
Butnariu, Dan; Klement, E. P. (1993). Triangular norm-based measures and games with fuzzy coalitions. Dordrecht: Kluwer Academic. p. 87. ISBN 0-7923-2369-6.

External links

Atom at The Encyclopedia of Mathematics

[1] Sierpinski, W. (1922). "Sur les fonctions d'ensemble additives et continues" (PDF). Fundamenta Mathematicae (in French). 3: 240–246.

[2] Fryszkowski, Andrzej (2005). Fixed Point Theory for Decomposable Sets (Topological Fixed Point Theory and Its Applications). New York: Springer. p. 39. ISBN 1-4020-2498-3.

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@@ Line 21: / Line 21: @@
 A measure <math> \mu </math> on a [[measurable space]] <math>(X, \Sigma)</math> is called '''atomic''' or '''purely atomic''' if every measurable set of positive measure contains an atom.
+== Discrete measures==
-The measure <math> \mu </math> is called discrete if it is a weighted sum of countably many Dirac measures, that is, there is a sequence <math> x_1,x_2,... </math> of points in <math> X </math>, and a sequence <math> c_1,c_2,... </math> of positive real numbers (the weights) such that <math> \mu=\sum_{k=1}^\infty c_k\delta_{x_k} </math>, which means that <math> \mu(A) = \sum_{k=1}^\infty c_k\delta_{x_k}(A) </math> for every <math> A\in\Sigma </math>.
+A <math> \sigma </math>-finite measure <math> \mu </math> is called discrete if it is atomic and the intersection of the atoms of any atomic class is non empty.
+It is equivalent to say that <math> \mu </math>  is the weighted sum of countably many Dirac measures, that is, there is a sequence <math> x_1,x_2,... </math> of points in <math> X </math>, and a sequence <math> c_1,c_2,... </math> of positive real numbers (the weights) such that <math> \mu=\sum_{k=1}^\infty c_k\delta_{x_k} </math>, which means that <math> \mu(A) = \sum_{k=1}^\infty c_k\delta_{x_k}(A) </math> for every <math> A\in\Sigma </math>. We can chose each point <math> x_n </math> to be a common point of the atoms
+in the <math> n </math>-th atomic class.
-A discrete measure is atomic but the inverse implication fails: take <math>X=[0,1]</math>, <math>\Sigma</math> the <math>\sigma</math>-algebra of countable and co-countable subsets,   <math> \mu=0 </math> in countable subsets and <math> \mu=1 </math> in co-countable subsets. Then there is a single atomic class, the one formed by the co-countable subsets, <math> \mu</math> is atomic but can't be put as a sum of Dirac measures.
+A discrete measure is atomic but the inverse implication fails: take <math>X=[0,1]</math>, <math>\Sigma</math> the <math>\sigma</math>-algebra of countable and co-countable subsets,   <math> \mu=0 </math> in countable subsets and <math> \mu=1 </math> in co-countable subsets. Then there is a single atomic class, the one formed by the co-countable subsets. The measure <math> \mu</math> is atomic but the intersection of the atoms in the unique atomic class is empty and <math> \mu </math> can't be put as a sum of Dirac measures.
 == Non-atomic measures==