Atom (measure theory)


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 non-atomic 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

 \mu (A) >0\,

and for any measurable subset B of A with

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

one has μ(B) = 0.

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.
  • Consider the Lebesgue measure on the real line. This measure has no atoms.

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) \geq b \geq0\,

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.

Sketch of proof of Sierpiński's theorem on non-atomic measures. A slightly stronger statement, which however makes the proof easier, is that if (X,Σ,μ) is a non-atomic measure space and μ(X) = c, there exists a function S:[0, c]\to\Sigma that is monotone with respect to inclusion, and a right-inverse to \mu:\Sigma\to[0,\,c]. That is, there exists a one-parameter family of measurable sets S(t) such that for all 0\leq t \leq t'\leq c

S(t)\subset S(t'),
\mu\left (S(t)\right)=t.

The proof easily follows from Zorn's lemma applied to the set of all monotone partial sections to μ :

\Gamma:=\{S:D\to\Sigma\; :\; D\subset[0,\,c],\, S\; \mathrm{ monotone }, \forall t\in D\; (\mu\left (S(t)\right)=t)\},

ordered by inclusion of graphs, \mathrm{graph}(S)\subset \mathrm{graph}(S'). 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

Notes

  1. ^ 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. 
  2. ^ 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. 

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