Atom (measure theory)
From Wikipedia, the free encyclopedia
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.
Formally, given a measure space (X,Σ) and a finite 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.
Contents |
[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.
- Consider the Lebesgue measure on the real line. This measure has no atoms.
[edit] Properties
A real-valued measurable function is constant almost everywhere on an atom (assuming that one uses the Borel algebra on the real numbers).
[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
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
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. One can prove 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
there exists a measurable subset B of A such that
This theorem is reminiscent of the intermediate value theorem for continuous functions.
[edit] See also
- atomic (order theory) — an analogous concept in order theory
