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| − | | + | {{MSC|28A}} |
| − | {{MSC|03E04}} ''in set theory''
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| − | {{MSC|28A}} ''in measure theory'' | |
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| | [[Category:Classical measure theory]] | | [[Category:Classical measure theory]] |
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| − | [[Category:Set theory]]
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| | {{TEX|done}} | | {{TEX|done}} |
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| − | ==Set theory==
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| − | A minimal non-zero element of a [[Partially ordered set|partially ordered set]] with a zero $0$, i.e. an element $p$ such that $0<x\leq p$ implies $x=p$.
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| − | ==Measure algebras==
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| − | For the definition and relevance in the theory of measure algebras we refer to [[Measure algebra]].
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| − | ==Classical measure theory==
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| | ===Definition=== | | ===Definition=== |
| − | Let $\mu$ be a (nonnegative) [[Measure|measure]] on a [[Algebra of sets|$\sigma$-algebra]] $\mathcal{S}$ of subsets of a set $X$. An element $a\in \mathcal{S}$ is called an ''atom'' of $\mu$ if
| + | An outer measure is a [[Set function|set function]] $\mu$ such that |
| − | *$\mu (A)>0$;
| + | * Its domain of definition is an hereditary [[Ring of sets|$\sigma$-ring]] (also called $\sigma$-ideal) of subsets of a given space $X$, i.e. a $\sigma$-ring $\mathcal{R}\subset \mathcal{P} (X)$ with the property that for every $E\in \mathcal{R}$ all subsets of $E$ belong to $\mathcal{R}$; |
| − | *For every $B\in \mathcal{S}$ with $B\subset A$ either $\mu (B)=0$ or $\mu (B)=\mu (A)$
| + | * Its range is the extended real half-line $[0, \infty$]; |
| − | (cp. with Section IV.9.8 of {{Cite|DS}} or {{Cite|Fe}}).
| + | * $\mu (\emptyset) =0$ and $\mu$ is ''$\sigma$-subadditive'' (also called ''countably subadditive''), i.e. for every countable family $\{E_i\}\subset \mathcal{R}$ the following inequality holds: |
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| − | '''Remark'''
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| − | 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 $\mu$.
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| − | ===Atomic measures===
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| − | 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 distribution|discrete distributions]].
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| − | ===Nonatomic measures===
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| − | A measure $\mu$ is called ''nonatomic'' it has no atoms.
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| − | ===Jordan decomposition===
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| − | 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 {{Cite|Fe}}), which is sometimes called a [[Continuous distribution|continuous distribution]]. This decomposition is sometimes called ''Jordan decomposition'', although several authors use this name in other contexts, see [[Jordan decomposition]].
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| − | ===Measures in the euclidean space===
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| − | If $\mu$ is a $\sigma$-finite measure on the [[Borel set|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 [[Delta-function|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
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| | \[ | | \[ |
| − | \mu (A) = \sum_{x_i\in A} \alpha_i \qquad \mbox{for every Borel set $A$}. | + | \mu \left(\bigcup_i E_i\right) \leq \sum_i \mu (E_i)\, . |
| | \] | | \] |
| − | ===Sierpinski's theorem===
| + | The most common outer measures are indeed defined on the full space $\mathcal{P} (X)$ of subsets of $X$. |
| − | A nonatomic measure takes a continuum of values. This is a corollary of the following Theorem due to Sierpinski (see {{Cite|Si}}):
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| − | '''Theorem'''
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| − | 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 $\mu (B) = b$.
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| − | ==Comment==
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| − | 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|Null object of a category]]).
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| − | ==References==
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| − | {|
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| − | |valign="top"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory", '''1''', Interscience (1958). {{MR|0117523}} {{ZBL|0635.47001}}
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| − | |-
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| − | |valign="top"|{{Ref|Fe}}|| W. Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its applications", '''2''', Wiley (1971).
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| − | |valign="top"|{{Ref|Lo}}|| M. Loève, "Probability theory", Princeton Univ. Press (1963). {{MR|0203748}} {{ZBL|0108.14202}}
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| − | |valign="top"|{{Ref|Si}}|| W. Sierpinski, "Sur le fonctions d'enseble additives et continuoes", '''3''', Fund. Math. (1922) pp. 240-246.
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| − | |}
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