Atom (measure theory)

Definition

Given a measurable space ( X , Σ ) and a measure μ on that space, a set A ⊂ X in Σ is called an atom if

and for any measurable subset B ⊂ A with

the set B has measure zero.

Examples

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

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

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 μ ( A ) > 0 , then for any real number b satisfying

there exists a measurable subset B of A such that

This theorem is due to Wacław Sierpiński. 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 ( X , Σ , μ ) is a non-atomic measure space and μ ( X ) = c , there exists a function S : [ 0 , c ] → Σ that is monotone with respect to inclusion, and a right-inverse to μ : Σ → [ 0 , c ] . That is, there exists a one-parameter family of measurable sets S(t) such that for all 0 ≤ t ≤ t ′ ≤ c

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

ordered by inclusion of graphs, g r a p h ( S ) ⊂ g r a p h ( S ′ ) . It's then standard to show that every chain in Γ has an upper bound in Γ , and that any maximal element of Γ has domain [ 0 , c ] , proving the claim.