Algebra of sets

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

Algebra of sets

Also called Boolean algebra or field of sets by some authors. A collection $\mathcal{A}$ of subsets of some set $X$ which contains the empty set and is closed under the set-theoretic operations of finite union, finite intersection and taking complements, i.e. such that

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

(see Section 4 of [Ha]). Indeed it is sufficient to assume that $\mathcal{A}$ satisfies the first two properties to conclude that also the third holds.

It follows easily that an algebra is also closed under the operation of taking differences. Algebras are special classes of rings of sets (also called Boolean rings). A ring of sets is a nonempty collection $\mathcal{R}$ of subsets of some set $X$ which is closed under the set-theoretic operations of finite union and difference. An algebra can be characterized as a ring containing the set $X$.

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 procedure to construct $\mathcal{A}$ is the following. Define $\mathcal{A}_0$ as the set of all elements of $\mathcal{B}$ and their complements. Define $\mathcal{A}_1$ as the elements which are intersections of finitely many elements of $\mathcal{A}_0$. $\mathcal{A}$ consists then of finite unions of arbitrary elements of $\mathcal{A}_1$.

$\sigma$-Algebra

An algebra of sets that is also closed under countable unions, cp. with Section 40 of [Ha] (also called Boolean $\sigma$-algebra or $\sigma$-field). Analogously one defines $\sigma$-rings (also called Boolean $\sigma$-rings) as rings of sets which are closed under countable unions (see Section 40 of [Ha]). $\sigma$-algebras of $X$ can be characterized as $\sigma$-rings which contain $X$.

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 $\mathcal{B}$ (also called Borel field generated by $\mathcal{B}$). A construction can be given using 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 (cp. with Exercise 9 of Section 5 in [Ha] where the same construction is outlined for $\sigma$-rings).

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 \[ \{x\in\mathbb R : a\leq x <b\} \qquad \mbox{where } -\infty \leq a <b\leq \infty \] 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 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