Complex measure

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

$\mu (A)>0$

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

$\mu (A)>\mu (B)$

one has $\mu (B)=0.$

Examples

• Consider the set X={1, 2, ..., 9, 10} and let the sigma-algebra $\mathrm{\Sigma }$ 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.

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 $A$ 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

$\mu (A)\ge b\ge 0$

there exists a measurable subset B of A such that

$\mu (B)=b.$

This theorem is due to Wacław Sierpiński. 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 $(X,\mathrm{\Sigma },\mu )$ is a non-atomic measure space and $\mu (X)=c$, there exists a function $S:[0,c]\to \mathrm{\Sigma }$ that is monotone with respect to inclusion, and a right-inverse to $\mu :\mathrm{\Sigma }\to [0,c]$. That is, there exists a one-parameter family of measurable sets S(t) such that for all $0\le t\le t^{\prime }\le c$

$S(t)\subset S(t^{\prime }),$

$\mu (S(t))=t.$

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

$\mathrm{\Gamma }:=\{S:D\to \mathrm{\Sigma }:D\subset [0,c],S\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{e},\mathrm{\forall }t\in D(\mu (S(t))=t)\},$

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

See also

• Atom (order theory) — an analogous concept in order theory
• Dirac delta function
• Elementary event, also known as an atomic event

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534