Atom (measure theory): Difference between revisions
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* Consider the set ''X''={1, 2, ..., 9, 10} and let the [[sigma-algebra]] <math>\Sigma</math> be the [[power set]] of ''X''. Define the measure <math>\mu</math> of a set to be its [[cardinality]], that is, the number of elements in the set. Then, each of the [[singleton (mathematics)|singleton]]s {''i''}, for ''i''=1,2, ..., 9, 10 is an atom. |
* Consider the set ''X''={1, 2, ..., 9, 10} and let the [[sigma-algebra]] <math>\Sigma</math> be the [[power set]] of ''X''. Define the measure <math>\mu</math> of a set to be its [[cardinality]], that is, the number of elements in the set. Then, each of the [[singleton (mathematics)|singleton]]s {''i''}, for ''i''=1,2, ..., 9, 10 is an atom. |
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* Consider the [[Lebesgue measure]] on the [[real line]]. This measure has no atoms. |
* Consider the [[Lebesgue measure]] on the [[real line]]. This measure has no atoms. |
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== Atomic measures== |
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A measure is called '''atomic''' or '''purely atomic''' if every measurable set of positive measure contains an atom. A (bounded, positive) measure <math> \mu </math> on a [[measurable space]] <math>(X, \Sigma)</math> is atomic if and only it is a weighted sum of countably many Dirac measures, that is, there is a sequence <math> x_1,x_2,... </math> of points in <math> X </math>, and a sequence <math> c_1,c_2,... </math> of positive real numbers (the weights) such that <math> \mu=\sum_{k=1}^\infty c_k\delta_{x_k} </math>, which means that |
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: <math> \mu(A) = \sum_{k=1}^\infty c_k\delta_{x_k}(A) </math> |
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for every <math> A\in\Sigma </math>. |
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== Non-atomic measures== |
== Non-atomic measures== |
Revision as of 05:29, 16 August 2020
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
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.
Examples
- 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
A measure is called atomic or purely atomic if every measurable set of positive measure contains an atom. A (bounded, positive) measure on a measurable space is atomic if and only it is a 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 .
Non-atomic measures
A measure which has no atoms is called non-atomic or diffuse. In other words, a measure is non-atomic if for any measurable set with there exists a measurable subset B of A such that
A non-atomic measure with at least one positive value has an infinite number of distinct values, as starting with a set A 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 μ is a non-atomic measure and A is a measurable set with then for any real number b satisfying
there exists a measurable subset B of A such that
This theorem is due to Wacław Sierpiński.[1][2] 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 S(t) 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 event
Notes
- ^ Sierpinski, W. (1922). "Sur les fonctions d'ensemble additives et continues" (PDF). Fundamenta Mathematicae (in French). 3: 240–246.
- ^ 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
- 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
- Atom at The Encyclopedia of Mathematics