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 "smaller" set of positive measure.

Formally, given a measure space $(X,Σ)$ and a finite measure $μ$ on that space, a set $A$ in $Σ$ is called an atom if

$\mu (A) >0\,$

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

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

one has $μ(B) = 0.$

1 Examples
2 Properties
3 Non-atomic measures
4 See also

[edit] Examples

Consider the set X={1, 2, ..., 9, 10} and let the sigma-algebra $Σ$ be the power set of X. Define the measure $μ$ 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.

[edit] Properties

A real-valued measurable function is constant almost everywhere on an atom (assuming that one uses the Borel algebra on the real numbers).

[edit] 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 $μ(A) > 0$ 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 $μ(A) > 0$ 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. One can prove that if μ is a non-atomic measure and A is a measurable set with $μ(A) > 0,$ then for any real number b satisfying

$\mu (A) > b >0\,$