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A collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r1200101.png" /> of subsets of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r1200102.png" /> satisfying:
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{{MSC|03E15|28A05}}
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[[Category:Descriptive set theory]]
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[[Category:Classical measure theory]]
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{{TEX|done}}
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r1200103.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r1200104.png" />;
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A collection $\mathcal{A}$ of subsets of a set $X$ satisfying:
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r1200105.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r1200106.png" />.
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i) $A\setminus B\in \mathcal{A}$ for every $A,B\in \mathcal{A}$;
  
It follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r1200107.png" /> is also closed under finite intersections, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r1200108.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r1200109.png" />, the ring of sets is an [[Algebra of sets|algebra of sets]].
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ii) $A\cup B\in \mathcal{A}$ for every $A,B\in \mathcal{A}$.
  
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r12001011.png" />-ring of sets is a ring of sets satisfying additionally
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It follows therefore that rings of sets are also closed under finite intersections. If the ring $\mathcal{A}$ contains $X$ then it is called an [[Algebra of sets|algebra of sets]].
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r12001012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r12001013.png" />, implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r12001014.png" />.
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A $\sigma$-ring is a ring which is closed under countable unions, i.e. such that
 
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\[
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r12001015.png" />-ring is closed under countable intersections. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r12001016.png" /> is a member of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r12001017.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r12001018.png" /> of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r12001019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r12001020.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r120/r120010/r12001022.png" />-algebra (cf. [[Additive class of sets|Additive class of sets]]; [[Algebra of sets|Algebra of sets]]).
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\bigcup_{i=1}^\infty A_i \in \mathcal{A} \qquad \mbox{whenever $\{A_i\}_{i\in \mathbb N}\subset \mathcal{A}$.}
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\]
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A $\sigma$-ring is therefore closed under countable intersections. If the $\sigma$-ring contains $X$, then it is called a [[Algebra of sets|$\sigma$-algebra]].  
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H.R. Pitt,  "Integration, measure and probability" , Oliver&amp;Boyd (1963) pp. 2–3</TD></TR></table>
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{|
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|valign="top"|{{Ref|Bo}}||      N. Bourbaki, "Elements of  mathematics. Integration",  Addison-Wesley    (1975) pp. Chapt.6;7;8  (Translated from French)  {{MR|0583191}}    {{ZBL|1116.28002}}  {{ZBL|1106.46005}}  {{ZBL|1106.46006}}    {{ZBL|1182.28002}}  {{ZBL|1182.28001}}  {{ZBL|1095.28002}}    {{ZBL|1095.28001}}  {{ZBL|0156.06001}}
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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|Ha}}|| P.R. Halmos,  "Measure theory", v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}}
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|-
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|valign="top"|{{Ref|Ne}}||  J. Neveu, "Mathematical foundations of the calculus of probability",  Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam  1965 {{MR|0198505}} {{ZBL|0137.1130}}
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|-
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|}

Revision as of 07:15, 19 September 2012

2020 Mathematics Subject Classification: Primary: 03E15 Secondary: 28A05 [MSN][ZBL]

A collection $\mathcal{A}$ of subsets of a set $X$ satisfying:

i) $A\setminus B\in \mathcal{A}$ for every $A,B\in \mathcal{A}$;

ii) $A\cup B\in \mathcal{A}$ for every $A,B\in \mathcal{A}$.

It follows therefore that rings of sets are also closed under finite intersections. If the ring $\mathcal{A}$ contains $X$ then it is called an algebra of sets.

A $\sigma$-ring is a ring which is closed under countable unions, i.e. such that \[ \bigcup_{i=1}^\infty A_i \in \mathcal{A} \qquad \mbox{whenever '"`UNIQ-MathJax10-QINU`"'.} \] A $\sigma$-ring is therefore closed under countable intersections. If the $\sigma$-ring contains $X$, then it is called a $\sigma$-algebra.

References

[Bo] N. Bourbaki, "Elements of mathematics. Integration", Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory", 1, Interscience (1958) MR0117523 Zbl 0635.47001
[Ha] P.R. Halmos, "Measure theory", v. Nostrand (1950) MR0033869 Zbl 0040.16802
[Ne] J. Neveu, "Mathematical foundations of the calculus of probability", Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam 1965 MR0198505 Zbl 0137.1130
How to Cite This Entry:
Ring of sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ring_of_sets&oldid=28038
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article