Atom (measure theory)

This article deals with the atom as a mathematical term. For further meanings of the word, see atom (disambiguation) .

A μ-atom , sometimes simply called an atom , is a term in measure theory , a branch of mathematics that deals with generalized length and volume concepts. A set with a positive (abstract) volume is a μ-atom, if each subset either has the same volume as the μ-atom or has the volume 0.

definition

A measure space is given . A set is called a μ-atom if and for each with that either or . ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle \mu (A)>0}$ ${\displaystyle B\in {\mathcal {A}}}$ ${\displaystyle B\subset A}$ ${\displaystyle \mu (B)=0}$ ${\displaystyle \mu (A\setminus B)=0}$

Related terminology

Atomless measure

A measure is called atomless if no -atoms exist. The Lebesgue measure is atomless. ${\displaystyle \mu }$ ${\displaystyle \mu }$

Purely atomic measure

A measure is called purely atomic if atoms exist and for the (finite or infinite) union of all atoms ${\displaystyle A_{n}}$

${\displaystyle A:=\bigcup _{n}A_{n}}$

holds that is. ${\displaystyle \mu (\Omega \setminus A)=0}$

example

If one chooses the basic space and chooses the power set as the σ-algebra and defines the measure on the point sets as the generator of the σ-algebra ${\displaystyle \Omega =\mathbb {N} =\{0,1,2,\dotsc \}}$ ${\displaystyle {\mathcal {P}}(\mathbb {N} )}$

${\displaystyle \mu (\{n\})={\begin{cases}0&{\text{ wenn }}n=0\\{\tfrac {1}{n}}&{\text{ wenn }}n\neq 0\end{cases}}}$, then:

• The crowd is not an atom, there . ${\displaystyle \{0\}}$ ${\displaystyle \mu }$ ${\displaystyle \mu (\{0\})=0}$
• All single element sets are atoms. ${\displaystyle \{n\},\,n>0}$
• Lots of things are for an atom. It is real, non-empty subsets are and and it is as well . So is an atom. ${\displaystyle \{0,n\}}$ ${\displaystyle n\geq 1}$ ${\displaystyle \mu }$ ${\displaystyle \mu (\{0,n\})={\tfrac {1}{n}}>0}$ ${\displaystyle \{0\}}$ ${\displaystyle \{n\}}$ ${\displaystyle \mu (\{0\})=0}$ ${\displaystyle \mu (\{0,n\}\setminus \{n\})=\mu (\{0\})=0}$ ${\displaystyle \{0,n\}}$
• The measure is purely atomic, since the union of the atoms with the quantity results and applies. If the atoms are chosen differently, their union can also result in the entire basic set. ${\displaystyle A_{n}=\{n\}}$ ${\displaystyle n\geq 1}$ ${\displaystyle A=\mathbb {N} \setminus \{0\}}$ ${\displaystyle \mu (\mathbb {N} \setminus A)=\mu (\{0\})=0}$

use

Atoms, for example, in the theory of probability used to criteria that specify under which from the convergence in probability of the almost sure convergence follows. If a sequence of random variables converges in probability against the random variable and if the basic space of the probability space can be represented as a disjoint union of atoms, then they almost certainly converge against . ${\displaystyle X}$ ${\displaystyle \Omega }$ ${\displaystyle X_{n}}$ ${\displaystyle X}$

Such a representation of the basic set as a disjoint union of atoms is always possible for probability spaces with at most a countable basic set.

literature

• Jürgen Elstrodt: Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .