Atom (measure theory)

Summary

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.

DefinitionEdit

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.

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.

ExamplesEdit

  • 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.

Atomic measuresEdit

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.[1] 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.

Discrete measuresEdit

A  -finite atomic measure   is called discrete if the intersection of the atoms of any atomic class is non empty. It is equivalent[2] 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 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.[3]

Non-atomic measuresEdit

A measure which has no atoms is called non-atomic measure or a diffuse measure. 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.[4][5] 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 alsoEdit

NotesEdit

  1. ^ "Analysis - Countable partition in atoms".
  2. ^ "Why must a discrete atomic measure admit a decomposition into Dirac measures? Moreover, what is "an atomic class"?".
  3. ^ Kadets, Vladimir (2018). A Course in Functional Analysis and Measure Theory. Switzerland: Springer. p. 45. ISBN 978-3-319-92003-0.
  4. ^ Sierpinski, W. (1922). "Sur les fonctions d'ensemble additives et continues" (PDF). Fundamenta Mathematicae (in French). 3: 240–246. doi:10.4064/fm-3-1-240-246.
  5. ^ Fryszkowski, Andrzej (2005). Fixed Point Theory for Decomposable Sets (Topological Fixed Point Theory and Its Applications). New York: Springer. p. 39. ISBN 1-4020-2498-3.

ReferencesEdit

  • Bruckner, Andrew M.; Bruckner, Judith B.; Thomson, Brian S. (1997). Real analysis. Upper Saddle River, N.J.: Prentice-Hall. p. 108. ISBN 0-13-458886-X.
  • Butnariu, Dan; Klement, E. P. (1993). Triangular norm-based measures and games with fuzzy coalitions. Dordrecht: Kluwer Academic. p. 87. ISBN 0-7923-2369-6.

External linksEdit

  • Atom at The Encyclopedia of Mathematics