that for every . We can chose each point to be a common point of the atoms in the -th atomic class. A discrete measure is atomic but the inverse implication fails: take , the -algebra of countable and co-countable subsets, in countable subsets and in co-countable subsets. Then there is a single atomic class, the one formed by the co-countable subsets. The measure is atomic but the intersection of the atoms in the unique atomic class is empty and can't be put as a sum of Dirac measures. If every atom is equivalent to a singleton, is discrete iff it is atomic. In this case the 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. Non-atomic measures A measure which has no atoms is called or a . In other words, a measure is non-atomic if for any measurable set with there exists a measurable subset of such that A non-atomic measure with at least one positive value has an infinite number of distinct values, as starting with a set 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 is a non-atomic measure and is a measurable set with then for any real number satisfying there exists a measurable subset of 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 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. See also Atom (order theory) — an analogous concept in order theory Dirac delta function Elementary event, also known as an atomic