Atom (measure theory)


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

Definition

Given a measurable space and a measure on that space, a set in is called an atom if
and for any measurable subset with
the set has measure zero.

Examples

A measure which has no atoms is called non-atomic or diffuse. In other words, a measure is non-atomic if for any measurable set with 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 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 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 is a non-atomic measure space and, there exists a function that is monotone with respect to inclusion, and a right-inverse to. That is, there exists a one-parameter family of measurable sets S such that for all
The proof easily follows from Zorn's lemma applied to the set of all monotone partial sections to :
ordered by inclusion of graphs, It's then standard to show that every chain in has an upper bound in, and that any maximal element of has domain proving the claim.