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.

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