 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 but positive measure. A measure which has no atoms is called nonatomic or atomless.
Contents
Definition
Given a measurable space (X,Σ) and a measure μ on that space, a set A in Σ is called an atom if
and for any measurable subset B of A with
one has μ(B) = 0.
Examples
 Consider the set X={1, 2, ..., 9, 10} and let the sigmaalgebra Σ be the power set of X. Define the measure μ 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.
Nonatomic measures
A measure which has no atoms is called nonatomic. In other words, a measure is nonatomic if for any measurable set A with μ(A) > 0 there exists a measurable subset B of A such that
A nonatomic measure with at least one positive value has an infinite number of distinct values, as starting with a set A with μ(A) > 0 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 nonatomic measures actually have a continuum of values. It can be proved that if μ is a nonatomic measure and A is a measurable set with μ(A) > 0, 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 nonatomic measures. A slightly stronger statement, which however makes the proof easier, is that if (X,Σ,μ) is a nonatomic measure space and μ(X) = c, there exists a function that is monotone with respect to inclusion, and a rightinverse to . That is, there exists a oneparameter 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 μ :
ordered by inclusion of graphs, It's then standard to show that every chain in Γ has a maximal element, and that any maximal element of Γ has domain [0,c], 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". Fundamenta Mathematicae 3: 240–246. http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3125.pdf.
 ^ Fryszkowski, Andrzej (2005). Fixed Point Theory for Decomposable Sets (Topological Fixed Point Theory and Its Applications). New York: Springer. p. 39. ISBN 1402024983.
References
 Bruckner, Andrew M.; Bruckner, Judith B.; Thomson, Brian S. (1997). Real analysis. Upper Saddle River, N.J.: PrenticeHall. p. 108. ISBN 013458886X.
 Butnariu, Dan; Klement, E. P. (1993). Triangular normbased measures and games with fuzzy coalitions. Dordrecht: Kluwer Academic. p. 87. ISBN 0792323696.
Categories: Measure theory
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