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. 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, i.e. . If is an atom, all the subsets in the -equivalence class of are atoms, and is called an atomic class. If is a -finite measure, there are countably many atomic classes. 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.