Algebra of sets

2020 Mathematics Subject Classification: Primary: 03A15 Secondary: 28A33 [MSN][ZBL]

Algebra of sets

A collection $\mathcal{A}$ of subsets of some set $X$ which contains the empty set and is closed under the set-theoretic operations of union, intersection and taking complements, i.e. such that

• $A\in\mathcal{A}\Rightarrow X\setminus A\in \mathcal{A}$;
• $A,B\in \mathcal{A}\Rightarrow A\cup B\in\mathcal{A}$;
• $A,B\in \mathcal{A}\Rightarrow A\cap B\in\mathcal{A}$.

Indeed it is sufficient to assume that $\mathcal{A}$ satisfies the first two properties to conclude that also the third holds.

The algebra generated by a family $\mathcal{B}$ of subsets of $X$ is defined as the smallest algebra $\mathcal{A}$ of subsets of $X$ containing $\mathcal{B}$. A simple inductive procedure allows to "construct" $\mathcal{A}$ as follows. $\mathcal{A}_0$ consists of all elements of $\mathcal{B}$ and their complements. For any $n\in\mathbb N\setminus \{0\}$ we define $\mathcal{A}_n$ as the collection of those sets which are finite unions or finite intersections of elements of $\mathcal{A}_{n-1}$. Then $\mathcal{A}=\bigcup_{n\in\mathbb N} \mathcal{A}_n$.

$\sigma$-Algebra

An algebra of sets that is also closed under countable unions. As a corollary a $\sigma$-algebra is also closed under countable intersections. As above, given a collection $\mathcal{B}$ of subsets of $X$, the $\sigma$-algebra generated by $\mathcal{B}$ is defined as the smallest $\sigma$-algebra of subsets of $X$ containing $\mathcal{B}$. The explicit construction given above for the algebra generated by $\mathcal{B}$ can be extended to $\sigma$-algebras with the aid of transfinite numbers. As above, $\mathcal{A}_0$consists of all elements of $\mathcal{B}$ and their complements. Given a countable ordinal $\alpha$, $\mathcal{A}_\alpha$ consists of those sets which are countable unions or countable intersections of elements belonging to \[ \bigcup_{\beta<\alpha} \mathcal{A}_\beta\, . \] $\mathcal{A}$ is the union of the classes $\mathcal{A}_\alpha$ where the index $\alpha$ runs over all countable ordinals.

Relations to measure theory

Algebras (respectively $\sigma$-algebras) are the natural domain of definition of finitely-additive ($\sigma$-additive) measures. Therefore $\sigma$-algebras play a central role in measure theory, see for instance Measure space.

According to the theorem of extension of measures, any $\sigma$-finite, $\sigma$-additive measure, defined on an algebra A, can be uniquely extended to a $\sigma$-additive measure defined on the $\sigma$-algebra generated by $A$.

Examples.

1) Let $X$ be an arbitrary set. The collection of finite subsets of $X$ and their complements is an algebra of sets (so-called finite-cofinite algebra). The collection of subsets of $X$ which are at most countable and of their complements is a $\sigma$-algebra (so-called countable-cocountable σ-algebra).

2) The collection of finite unions of intervals of the type \[ \{x\in\mathbb R : a\leq x <b\} \qquad \mbox{where '"`UNIQ-MathJax49-QINU`"'} \] is an algebra.

3) If $X$ is a topological space, the elements of the $\sigma$-algebra generated by the open sets are called Borel sets.

4) The Lebesgue measurable sets of $\mathbb R^k$ form a $\sigma$ algebra (so-called Lebesgue σ-algebra, see Lebesgue measure).

5) Let $T$ be an arbitrary set and consider $X = \mathbb R^T$ (i.e. the set of all real-valued functions on $\mathbb R$). Let $A$ be the class of sets of the type \[ \{\omega\in \mathbb R^T: (\omega (t_1), \ldots,\omega t_k)\in E\} \] where $k$ is an arbitrary natural number, $E$ an arbitrary Borel subset of $\mathbb R^k$ and $t_1,\ldots, t_k$ an arbitrary collection of distinct elements of $T$. $A$ is an algebra of subsets of $\mathbb R^T$ (so-called cylindrical algebra). In the theory of random processes a probability measure is often originally defined only on an algebra of this type, and then subsequently extended to the $\sigma$-algebra generated by $A$.

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
[Ha]	P.R. Halmos, "Measure theory", v. Nostrand (1950) MR0033869 Zbl 0040.16802
[Ne]	J. Neveu, "Bases mathématiques du calcul des probabilités", Masson (1970)