Ring of sets

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

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

i) $\emptyset\in \mathcal{A}$;

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

iii) $A\cup B\in \mathcal{A}$ 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 \[ \bigcup_{i=1}^\infty A_i \in \mathcal{A} \qquad \mbox{whenever } \{A_i\}_{i\in \mathbb N}\subset \mathcal{A}\, . \] A $\sigma$-ring is therefore closed under countable intersections. If the $\sigma$-ring contains $X$, then it is called a $\sigma$-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