Atom (measure theory)

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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.

Definition

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

${\displaystyle \mu (A)>0}$

and for any measurable subset ${\displaystyle B\subset A}$ with

${\displaystyle \mu (B)<\mu (A)}$

the set ${\displaystyle B}$ has measure zero.

Examples

• Consider the set X={1, 2, ..., 9, 10} and let the sigma-algebra ${\displaystyle \Sigma }$ be the power set of X. Define the measure ${\displaystyle \mu }$ 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.

Non-atomic measures

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

${\displaystyle \mu (A)>\mu (B)>0.\,}$

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

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

such that

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

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 ${\displaystyle \mu (A)>0,}$ then for any real number b satisfying

${\displaystyle \mu (A)\geq b\geq 0\,}$

there exists a measurable subset B of A such that

${\displaystyle \mu (B)=b.\,}$

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 ${\displaystyle (X,\Sigma ,\mu )}$ is a non-atomic measure space and ${\displaystyle \mu (X)=c}$, there exists a function ${\displaystyle S:[0,c]\to \Sigma }$ that is monotone with respect to inclusion, and a right-inverse to ${\displaystyle \mu :\Sigma \to [0,\,c]}$. That is, there exists a one-parameter family of measurable sets S(t) such that for all ${\displaystyle 0\leq t\leq t'\leq c}$

${\displaystyle S(t)\subset S(t'),}$
${\displaystyle \mu \left(S(t)\right)=t.}$

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

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

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

Notes

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.

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.