Difference between revisions of "Ring of sets"
From Encyclopedia of Mathematics
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| − | + | {{MSC|03E15|28A05}} | |
| + | [[Category:Descriptive set theory]] | ||
| + | [[Category:Classical measure theory]] | ||
| + | {{TEX|done}} | ||
| − | + | 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|algebra of sets]]. | |
| − | a | + | A $\sigma$-ring is a ring which is closed under countable unions, i.e. such that |
| − | + | \[ | |
| − | A | + | \bigcup_{i=1}^\infty A_i \in \mathcal{A} \qquad \mbox{whenever $\{A_i\}_{i\in \mathbb N}\subset \mathcal{A}$.} |
| + | \] | ||
| + | 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==== | ||
| − | + | {| | |
| + | |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}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory", '''1''', Interscience (1958) {{MR|0117523}} {{ZBL|0635.47001}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Measure theory", v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}} | ||
| + | |- | ||
| + | |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}} | ||
| + | |- | ||
| + | |} | ||
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=14767
Ring of sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ring_of_sets&oldid=14767
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article