In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no "smaller" set of positive measure. Formally, given a measure space <math>(X, \Sigma)</math> and a finite measure <math>\mu</math> on that space, a set <math>A</math> in <math>\Sigma</math> is called an atom if
- <math> \mu (A) >0\, </math>
and for any measurable subset <math>B</math> of <math>A</math> with
- <math> \mu(A) > \mu (B) \, </math>
one has <math> \mu(B)=0.</math>
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Examples
- Consider the set X={1, 2, ..., 9, 10} and let the sigma-algebra <math>\Sigma</math> be the power set of X. Define the measure <math>\mu</math> 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.
Properties
A real-valued measurable function is constant almost everywhere on an atom (assuming that one uses the Borel algebra on the real numbers).
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 <math>A</math> with <math> \mu (A) >0</math> there exists a measurable subset B of A such that
- <math> \mu(A) > \mu (B) > 0. \, </math>
A non-atomic measure with at least one positive value has an infinite number of distinct values, as starting with a set A with <math> \mu (A) >0</math> one can construct a decreasing sequence of measurable sets
- <math>A=A_1\supset A_2 \supset A_3 \supset \cdots</math>
such that
- <math>\mu(A)=\mu(A_1) > \mu(A_2) > \mu(A_3) > \cdots > 0. </math>
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. One can prove that if μ is a non-atomic measure and A is a measurable set with <math>\mu (A) >0,</math> then for any real number b satisfying
- <math>\mu (A) > b >0\, </math>
there exists a measurable subset B of A such that
- <math>\mu(B)=b.\,</math>
This theorem is reminiscent of the intermediate value theorem for continuous functions.
See also
- atomic (order theory) — an analogous concept in order theory
