Atom (measure theory)

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

μ (A) >0

and for any measurable subset B of A with

μ(A) > μ (B)

one has μ(B) = 0.

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.

Non-atomic measures

μ(A) > μ (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 ⊃ A ₂ ⊃ A ₃ ⊃ …

such that

μ(A) = μ(A ₁ ) > μ(A ₂ ) > μ(A ₃ ) > … > 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

μ (A) > b >0

there exists a measurable subset B of A such that

μ(B) = b.

This theorem is reminescent of the Intermediate value theorem for continous functions.

See also

• Atomic (order theory) - an analogous concept in order theory