2020 Mathematics Subject Classification: Primary: 03E15 Secondary: 28A05 [MSN][ZBL]
Algebra of sets[edit]
Also called Boolean algebra or field of sets by some authors. A collection of subsets of some set 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
(see Section 4 of [Ha]). Indeed it is sufficient to assume that 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 of subsets of some set which is closed under the set-theoretic operations of finite union and difference. An algebra can be characterized as a ring containing the set .
The algebra generated by a family of subsets of is defined as the smallest algebra of subsets
of containing . A simple procedure to construct is the following. Define
as the set of all elements of and their complements. Define as the elements which are intersections of finitely many elements of . consists then of finite unions of arbitrary elements of .
-Algebra[edit]
An algebra of sets that is also closed under countable unions, cp. with Section 40 of [Ha] (also called Boolean -algebra or -field). Analogously one defines -rings (also called Boolean -rings) as rings of sets which are closed under countable unions (see Section 40 of [Ha]). -algebras of can be characterized as -rings which contain .
As a corollary a -algebra is also closed
under countable intersections. As above, given a collection of subsets of , the -algebra generated
by is defined as the smallest -algebra of subsets of containing (also called Borel field generated by ). A
construction can be given using
transfinite numbers. As above, consists of all elements of and their complements.
Given a countable ordinal , consists of those sets which are countable unions or countable intersections
of elements belonging to
is the union of the classes where the index runs over all countable ordinals
(cp. with Exercise 9 of Section 5 in [Ha] where the same construction is outlined for -rings).
Relations to measure theory[edit]
Algebras (respectively -algebras) are the natural domain of definition of finitely-additive (-additive) measures.
Therefore -algebras play a central role in measure theory, see for instance Measure space.
According to the theorem of extension of measures, any -finite, -additive measure, defined on an algebra , can be uniquely extended to a -additive measure defined on the -algebra generated by .
Examples.[edit]
1) Let be an arbitrary set. The collection of finite subsets of and their complements is an algebra of sets (so-called finite-cofinite algebra). The collection of subsets
of which are at most countable and of their complements is a -algebra (so-called countable-cocountable σ-algebra).
2) The collection of finite unions of intervals of the type
is an algebra.
3) If is a topological space, the elements of the -algebra generated by the open sets are called Borel sets.
4) The Lebesgue measurable sets of form a -algebra (so-called Lebesgue σ-algebra, see Lebesgue measure).
5) Let be an arbitrary set and consider (i.e. the set of all real-valued functions on ).
Let be the class of sets of the type
where is an arbitrary natural number, an arbitrary Borel subset of and
an arbitrary collection of distinct elements of . is an algebra of subsets of (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 -algebra generated by .
References[edit]
[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
|