In mathematics, more precisely in measure theory, an atom is a measurable set that has positive measure and contains no set of smaller positive measures. A measure that has no atoms is called non-atomic or atomless.

Definition

Given a measurable space and a measure is called an atom.

The equivalence class is the symmetric difference is called an atomic class.-finite measure, there are countably many atomic classes.

Examples

• Consider the set X = {1, 2, ..., 9, 10} and let the sigma-algebra be the power set of X of a set to be its cardinality, that is, the number of elements in the set. Then, each of the singletonss {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

on a measurable space is called atomic or purely atomic if every measurable set of positive measure contains an atom. This is equivalent to say that there is a countable formed by atoms up to a null set. denotes the counting measure. This space is atomic, with all atoms being the singletons since the countable union of singletons is a countable set, and the uncountability of the real numbers-finite spaces follows from the proof for finite measure spaces by observing that the countable union of countable unions is again a countable union, and that the countable unions of null sets are null.

Discrete measures

is called discrete if the intersection of the atoms of any atomic class is non empty. It is equivalent-th atomic class.

can't be put as a sum of Dirac measures.

above are the atomic singletons, so they are unique. Any finite measure in a separable metric space provided with the Borel sets satisfies this condition.

Non-atomic measures

A measure which has no atoms is called or a

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

This theorem is due to Wacław Sierpiński. It is reminiscent of the intermediate value theorem for continuous functions.

Sketch of proof The proof easily follows from Zorn's lemma 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

References

External links

• Atom at The Encyclopedia of Mathematics