User:Camillo.delellis/sandbox

2020 Mathematics Subject Classification: Primary: 03E04 [MSN][ZBL] in set theory

2020 Mathematics Subject Classification: Primary: 28A [MSN][ZBL] in measure theory

Set theory

A minimal non-zero element of a partially ordered set with a zero $0$, i.e. an element $p$ such that $0<x\leq p$ implies $x=p$.

Measure algebras

For the definition and relevance in the theory of measure algebras we refer to Measure algebra.

Classical measure theory

Definition

Let $\mu$ be a (nonnegative) measure on a $\sigma$-algebra $\mathcal{S}$ of subsets of a set $X$. An element $a\in \mathcal{S}$ is called an atom of $\mu$ if

• $\mu (A)>0$;
• For every $B\in \mathcal{S}$ with $B\subset A$ either $\mu (B)=0$ or $\mu (B)=\mu (A)$

(cp. with Section IV.9.8 of [DS] or [Fe]).

Remark If we denote by $\mathcal{N}$ the null sets and consider the standard quotient measure algebra $(\mathcal{S}/\mathcal{N}, \mu)$, then any atom of such quotient measure algebra corresponds to an equivalence class of atoms of $\mu$.

Atomic measures

A measure $\mu$ is called atomic if there is a partition of $X$ into countably many elements of $\mathcal{A}$ which are either atoms or null sets. An atomic probability neasure is often called atomic distribution. Examples of atomic distributions are the discrete distributions.

Nonatomic measures

A measure $\mu$ is called nonatomic it has no atoms.

Jordan decomposition

If $\mu$ is $\sigma$-finite, it is possible to decompose $\mu$ as $\mu_a+\mu_{na}$, where $\mu_a$ is an atomic measure and $\mu_{na}$ is a nonatomic measure. In case $\mu$ is a probability measure, this means that $\mu$ can be written as $p \mu_a + (1-p) \mu_{na}$, where $p\in [0,1]$, $\mu_a$ is an atomic probability measure and $\mu_{na}$ a nonatomic probability measure (see [Fe]), which is sometimes called a continuous distribution. This decomposition is sometimes called Jordan decomposition, although several authors use this name in other contexts, see Jordan decomposition.

Measures in the euclidean space

If $\mu$ is a $\sigma$-finite measure on the Borel $\sigma$-algebra of $\mathbb R^n$, then it is easy to show that, for any atom $B$ of $\mu$ there is a point $x\in B$ with the property that $\mu (B) = \mu (\{x\})$. Thus such a measure is atomic if and only if it is the countable sum of Dirac deltas, i.e. if there is an (at most) countable set $\{x_i\}\subset \mathbb R^n$ and an (at most) countable set $\{\alpha_i\}\subset ]0, \infty[$ with the property that \[ \mu (A) = \sum_{x_i\in A} \alpha_i \qquad \mbox{for every Borel set '"`UNIQ-MathJax46-QINU`"'}. \]

Sierpinski's theorem

A nonatomic measure takes a continuum of values. This is a corollary of the following Theorem due to Sierpinski (see [Si]):

Theorem If $\mu$ is a nonatomic measure on a $\sigma$-algebra $\mathcal{A}$ and $A\in \mathcal{A}$ an element such that $\mu (A)>0$, then for every $b\in [0, \mu (B)]$ there is an element $B\in \mathcal{A}$ with $B\subset A$ and $\mu (B) = b$.

Comment

By a natural extension of meaning, the term atom is also used for an object of a category having no subobjects other than itself and the null subobject (cf. Null object of a category).

References

[DS]	N. Dunford, J.T. Schwartz, "Linear operators. General theory", 1, Interscience (1958). MR0117523 Zbl 0635.47001
[Fe]	"An introduction to probability theory and its applications", 2, Wiley (1971).
[Lo]	M. Loève, "Probability theory", Princeton Univ. Press (1963). MR0203748 Zbl 0108.14202
[Si]	W. Sierpinski, "Sur le fonctions d'enseble additives et continuoes", 3, Fund. Math. (1922) pp. 240-246.