A measure that has no atoms is called non-atomic or atomless.
is the symmetric difference operator.
-finite measure, there are countably many atomic classes.
This is equivalent to say that there is a countable partition of
formed by atoms up to a null set.
This space is atomic, with all atoms being the singletons, yet the space is not able to be partitioned into the disjoint union of countably many disjoint atoms,
since the countable union of singletons is a countable set, and the uncountability of the real numbers shows that the complement
-measure would be infinite, in contradiction to it being a null set.
-finite spaces follows from the proof for finite measure spaces by observing that the countable union of countable unions is again a countable union, and that the countable unions of null sets are null.
is the weighted sum of countably many Dirac measures, that is, there is a sequence
of positive real numbers (the weights) such that
A discrete measure is atomic but the inverse implication fails: take
-algebra of countable and co-countable subsets,
Then there is a single atomic class, the one formed by the co-countable subsets.
can't be put as a sum of Dirac measures.
If every atom is equivalent to a singleton, then
above are the atomic singletons, so they are unique.
Any finite measure in a separable metric space provided with the Borel sets satisfies this condition.
is non-atomic if for any measurable set
A non-atomic measure with at least one positive value has an infinite number of distinct values, as starting with a set
one can construct a decreasing sequence of measurable sets
It turns out that non-atomic measures actually have a continuum of values.
This theorem is due to Wacław Sierpiński.
[6][7] 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
that is monotone with respect to inclusion, and a right-inverse to
That is, there exists a one-parameter family of measurable sets
The proof easily follows from Zorn's lemma applied to the set of all monotone partial sections to
ordered by inclusion of graphs,