# Atom (measure theory)

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 and a measure on that space, a set in is called an **atom** if

and for any measurable subset with

the set has measure zero.

## Examples

- Consider the set
*X*={1, 2, ..., 9, 10} and let the sigma-algebra be the power set of*X*. Define the measure 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 with there exists a measurable subset *B* of *A* such that

A non-atomic measure with at least one positive value has an infinite number of distinct values, as starting with a set *A* with one can construct a decreasing sequence of measurable sets

such that

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 then for any real number *b* satisfying

there exists a measurable subset *B* of *A* such that

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 is a non-atomic measure space and , there exists a function that is monotone with respect to inclusion, and a right-inverse to . That is, there exists a one-parameter family of measurable sets S(t) such that for all

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

ordered by inclusion of graphs, It's then standard to show that every chain in has an upper bound in , and that any maximal element of has domain 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**

## 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