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 "smaller" set of positive measure. A measure which has no atoms is called non-atomic.

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[edit] Definition

Given a measurable space (X,Σ) and a finite measure μ on that space, a set A in Σ is called an atom if

\mu (A) >0\,

and for any measurable subset B of A with

\mu(A) > \mu (B) \,

one has μ(B) = 0.

[edit] Examples

  • Consider the set X={1, 2, ..., 9, 10} and let the sigma-algebra 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.

[edit] Non-atomic measures

A measure which has no atoms is called non-atomic. In other words, a measure is non-atomic if for any measurable set A with μ(A) > 0 there exists a measurable subset B of A such that

\mu(A) > \mu (B) > 0. \,

A non-atomic 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

A=A_1\supset A_2 \supset A_3 \supset \cdots

such that

\mu(A)=\mu(A_1) > \mu(A_2) > \mu(A_3) > \cdots > 0.

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 μ(A) > 0, then for any real number b satisfying

\mu (A) > b >0\,

there exists a measurable subset B of A such that

\mu(B)=b.\,

This theorem is due to Wacław Sierpiński. [1] [2] It is reminiscent of the intermediate value theorem for continuous functions.

[edit] See also

[edit] References

  1. ^ W. Sierpinski. Sur les fonctions d'ensemble additives et continues. Fund. Math., 3:240-246, 1922.
  2. ^ Fryszkowski, Andrzej. Fixed Point Theory for Decomposable Sets (Topological Fixed Point Theory and Its Applications). Springer, page 39. ISBN 1-4020-2498-3. 
  • Bruckner, Andrew M.; Bruckner, Judith B.; Thomson, Brian S. (1997). Real analysis. Upper Saddle River, N.J.: Prentice-Hall, page 108. ISBN 0-13-458886-X. 
  • Butnariu, Dan; Klement, E. P. (1993). Triangular norm-based measures and games with fuzzy coalitions. Dordrecht: Kluwer Academic, page 87. ISBN 0-7923-2369-6. 
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