Өлшем теориясындағы атомдық жиынтар және атомдық емес өлшемдер

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

Consider the set X = {1, 2, , 9, 10} and let the sigma algebra be the power set of X. Define the measure 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.

A finite measure on a measurable space is called atomic or purely atomic if every measurable set of positive measure contains an atom. This is equivalent to say that there is a countable partition of formed by atoms up to a null set. The assumption of finitude is essential. Consider otherwise the space where denotes the counting measure. 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, and a null set since the countable union of singletons is a countable set, and the uncountability of the real numbers shows that the complement would have to be uncountable, hence its measure would be infinite, in contradiction to it being a null set. The validity of the result for 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.

A finite atomic measure is called discrete if the intersection of the atoms of any atomic class is non empty. It is equivalent to say that is the weighted sum of countably many Dirac measures, that is, there is a sequence of points in , and a sequence of positive real numbers (the weights) such that , which means that for every We can choose 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, then 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.