atom (measure theory)
   HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, more precisely in
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
, an atom is a measurable set that has positive measure and contains no set of smaller positive measures. A measure that has no atoms is called non-atomic or atomless.


Definition

Given a
measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. It captures and generalises intuitive notions such as length, area, an ...
(X, \Sigma) and a measure \mu on that space, a set A\subset X in \Sigma is called an atom if \mu(A) > 0 and for any measurable subset B \subseteq A, either \mu(B) = 0 or \mu(B)=\mu(A). The
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
of A is defined by := \, where \Delta is the
symmetric difference In mathematics, the symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and ...
operator. If A is an atom then all the subsets in /math> are atoms and /math> is called an atomic class. If \mu is a \sigma-finite measure, there are countably many atomic classes.


Examples

* Consider the set ''X'' = and let the sigma-algebra \Sigma be the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of ''X''. Define the measure \mu of a set to be its
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
, that is, the number of elements in the set. Then, each of the singletons , for ''i'' = 1, 2, ..., 9, 10 is an atom. * Consider the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
on the
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
. This measure has no atoms.


Atomic measures

A \sigma-finite measure \mu on a
measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. It captures and generalises intuitive notions such as length, area, an ...
(X, \Sigma) is called atomic or purely atomic if every measurable set of positive measure contains an atom. This is equivalent to say that there is a
countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
partition of X formed by atoms up to a null set. The assumption of \sigma-finitude is essential. Consider otherwise the space (\mathbb,\mathcal(\Reals),\nu) where \nu denotes the
counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinit ...
. This space is atomic, with all atoms being the singletons, yet the space is not able to be partitioned into the disjoint union of countably many disjoint atoms, \bigcup_^\infty A_n and a null set N since the countable union of singletons is a countable set, and the uncountability of the real numbers shows that the complement N = \mathbb \setminus \bigcup_^\infty A_n would have to be uncountable, hence its \nu-measure would be infinite, in contradiction to it being a null set. The validity of the result for \sigma-finite spaces follows from the proof for finite measure spaces by observing that the countable union of countable unions is again a countable union, and that the countable unions of null sets are null.


Discrete measures

A \sigma-finite atomic measure \mu is called discrete if the intersection of the atoms of any atomic class is non empty. It is equivalent to say that \mu is the weighted sum of countably many Dirac measures, that is, there is a sequence x_1,x_2,... of points in X , and a sequence c_1,c_2,... of positive real numbers (the weights) such that \mu=\sum_^\infty c_k\delta_ , which means that \mu(A) = \sum_^\infty c_k\delta_(A) for every A\in\Sigma . We can choose each point x_k to be a common point of the atoms in the k -th atomic class. A discrete measure is atomic but the inverse implication fails: take X= ,1/math>, \Sigma the \sigma-algebra of countable and co-countable subsets, \mu=0 in countable subsets and \mu=1 in co-countable subsets. Then there is a single atomic class, the one formed by the co-countable subsets. The measure \mu is atomic but the intersection of the atoms in the unique atomic class is empty and \mu can't be put as a sum of Dirac measures. If every atom is equivalent to a singleton, then \mu is discrete iff it is atomic. In this case the x_k above are the atomic singletons, so they are unique. Any finite measure in a separable metric space provided with the Borel sets satisfies this condition.


Non-atomic measures

A measure which has no atoms is called or a . In other words, a measure \mu is non-atomic if for any measurable set A with \mu(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 \mu(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 \mu is a non-atomic measure and A is a measurable set with \mu(A) > 0, then for any real number b satisfying \mu(A) \geq b \geq 0 there exists a measurable subset B of A such that \mu(B) = b. This theorem is due to
Wacław Sierpiński Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions ...
. It is reminiscent of the
intermediate value theorem In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two imp ...
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, \Sigma, \mu) is a non-atomic measure space and \mu(X) = c, there exists a function S :
, c The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
\to \Sigma that is monotone with respect to inclusion, and a right-inverse to \mu : \Sigma \to
, c The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
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) \subseteq S(t'), \mu\left (S(t)\right)=t. The proof easily follows from
Zorn's lemma Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least on ...
applied to the set of all monotone partial sections to \mu : \Gamma: = \, ordered by inclusion of graphs, \mathrm(S) \subseteq \mathrm(S'). It's then standard to show that every chain in \Gamma has an upper bound in \Gamma, and that any maximal element of \Gamma has domain
, c The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
proving the claim.


See also

*
Atom (order theory) In the mathematical field of order theory, an element ''a'' of a partially ordered set with least element 0 is an atom if 0 < ''a'' and there is no ''x'' such that 0 < ''x'' < ''a''. Equivalently, one may define an atom to be an element that is ...
— an analogous concept in order theory *
Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
*
Elementary event In probability theory, an elementary event, also called an atomic event or sample point, is an event which contains only a single outcome in the sample space. Using set theory terminology, an elementary event is a singleton. Elementary events ...
, also known as an atomic event


Notes


References

* * * *


External links


Atom
at The Encyclopedia of Mathematics {{Measure theory Measure theory