Atom (measure theory)

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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.

Definition[edit]

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.

Examples[edit]

  • 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 measures[edit]

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.

Discrete measures[edit]

An 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 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.[1]

Non-atomic measures[edit]

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.[2][3] 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[edit]

Notes[edit]

  1. ^ Kadets, Vladimir (2018). A Course in Functional Analysis and Measure Theory. Switzerland: Springer. p. 45. ISBN 978-3-319-92003-0.
  2. ^ Sierpinski, W. (1922). "Sur les fonctions d'ensemble additives et continues" (PDF). Fundamenta Mathematicae (in French). 3: 240–246.
  3. ^ 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.

References[edit]

  • 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 links[edit]

  • Atom at The Encyclopedia of Mathematics