sigma-algebra
By Jimmy Tseng
σ-algebra, a mathematical concept, is a collection of subsets of a given set. Alternate names for this concept are sigma-algebra, σ-field, and sigma-field (σ is the Greek letter pronounced like sigma). σ-algebra is defined in analysis, one of the major branches of modern mathematics. It is a key concept necessary for the definition of measure, which is itself a key concept in analysis. Another branch of mathematics, probability, employs σ-algebra as a key concept because of its use of measure. Mathematicians who research and study probability often refer to σ-algebras as σ-fields.
Introduction to Algebras Over a Set
An algebra over a given set, X, is a non-empty collection of subsets of X that has three ways of combining subsets to obtain a new subset in the algebra. The first two ways are set intersections, denoted by ∩, and set unions, denoted by U. Given any finite sequence of elements of the algebra, their intersection and union are in the algebra; these rules are called closed under finite intersections and finite unions respectively. The third way of obtaining a new subset is by complements. For a subset E of X, its complement in X is the subset of X that contains all elements of X not in E. The complement of E is denoted below by E’, but is traditionally denoted by an E with a superscript c or by X\E. Note that the union and intersection are operations with two or more finite elements. The complement operation is on one element.
The reader should be aware that algebras over sets are different from the traditional algebras defined in the major branch of mathematics called algebra. One way to discern the difference between the two concepts of algebra is to note that inverses do not exist in an algebra over a set, but do exist in a traditional algebra. Consider the intersection operation; pretend for the moment that it is only a binary (2-element) operation. Let E be a subset of X. Then E ∩ X = E and X ∩ E = E. Hence, X is an identity for the intersection operation since any subset, E, intersected with X gives E. To find an inverse for E, one must find a subset of X such that it intersect E gives X. Clearly the only way this can occur is if E = X. Then X ∩ X = X. Otherwise, inverses do not exist. A similar argument can be used for union. Consequently, an algebra over a set and a traditional algebra share only a name and some loose identification, but are otherwise unrelated.
The Definition of Algebra over a Set
The succinct definition of an algebra over a set is as follows. Let X be a non-empty set. An algebra over X is a non-empty collection, A, of subsets of X that is closed under complements and finite unions.
Let E1, …, En be elements of A. Then E1’, …, En’ are also elements of A. And E1’ U … U En’ is an element of A. Hence, by de Morgan’s law, E1 ∩ … ∩ En is an element of A. Note that the union and intersection are both finite.
σ-algebra is a Generalization of Algebra Over a Set
σ-algebra is a generalization of an algebra over a set. Simply put, a σ-algebra is an algebra over a set X that is closed under countably infinite unions: for any countable infinite sequence of subsets in the σ-algebra, its union is also in the σ-algebra. To handle any finite sequence, one can append a countably infinite sequence of empty sets to the end of the finite sequence to obtain a countably infinite sequence whose union is the same as that of the finite sequence. Hence, "closed under countably infinite unions" includes "closed under finite unions."
The Definition of σ-algebra
Let X be a non-empty set. A σ-algebra over a set X is a non-empty collection, A, of subsets of X that is closed under complements and countably infinite unions.
The terms σ-algebra, σ-field, sigma-algebra, and sigma-field are interchangeable.
Elementary Properties of σ-algebras
Let X be a non-empty set. Let A be a σ-algebra over X. Since X is non-empty, X has at least two distinct subsets: X and ø (the empty set). Since A is non-empty, A has at least one element. Call this element E. Then E’ is in A. Since E U E’ = X, X is in A. Since X’ = ø, then ø is in A. Consequently, X and ø are in every σ-algebra over X.
Let {En} be a countably infinite sequence of elements of A. Since A is closed under complements, the countably infinite sequence {En’} is composed of elements in A. Its union, E1’ U E2’ U … U En’ …, is in A. Applying de Morgan’s law to that union, one obtains E1 ∩ E2 ∩ … ∩ En …. Hence, A is closed under countably infinite intersections too.
Examples of σ-algebras
Let X be a non-empty set. Then the collection {X, ø} is a σ-algebra since X’ = ø, ø’ = X, and X U ø = X. This σ-algebra is often called the trivial σ-algebra over X. The trivial σ-algebra is important since it is the σ-algebra over X that has the least number of elements, and it is a subset of any other σ-algebra over X.
Another σ-algebra, the power set of X denoted by P(X), is also important. P(X) is defined as the collection of all subsets of X. If E is in P(X), then so is E’ since E’ is a subset of X. Also any countable union of subsets of X is a subset of X. Hence, P(X) is a σ-algebra. Since every element of any σ-algebra over X is a subset of X, it must also be in P(X). Thus P(X) contains all σ-algebras over X.
Constructing σ-algebras
After a general definition is given, the neophyte mathematics student often has difficulty applying the definition to produce concrete examples. The usual tactic that the mathematician employs is the attempt to generate an object that satisfies the general definition by considering a few elements. The tactic works as follows. Let X be a non-empty set. Let E be a non-empty subset of P(X), the power set of X. Consider the collection of σ-algebras that contain E. It must be non-empty since P(X) is in it. Take the intersection over this collection of σ-algebras. If this intersection is also a unique σ-algebra over X, then this σ-algebra over X is the smallest σ-algebra that contains E. Denote this intersection by M(E).
To show that M(E) is a σ-algebra over X, one must check that M(E) is closed under complements and countably infinite unions. Let F be in M(E). Then F is in every σ-algebra over X containing E. Hence, F’ is in every σ-algebra over X containing E. Thus, F’ is in M(E). Let {Fn} be a countably infinite sequence of elements of M(E). Then {Fn} is countably infinite sequence of elements in every σ-algebra over X containing E. Hence, the union is in every σ-algebra over X containing E. Thus, the union is in M(E). Therefore, M(E) is a σ-algebra over X. And it contains E.
To show that M(E) is unique, one can assume that another σ-algebra over X containing E is the smallest. Let N(E) be that σ-algebra. Now N(E) is contained in M(E) since N(E) is the smallest. But, M(E) is the intersection of all σ-algebras over X that contain E. Hence, M(E) is contained in every such σ-algebra, including N(E). Thus, N(E) = M(E). Therefore, M(E) is unique.
M(E) is called the σ-algebra over X generated by E, the given subset of P(X). Note that, in this section, E is not a subset of X, but a subset of P(X). Hence, E is a collection of subsets of X.
An Example of Generated σ-algebras
A final example of σ-algebras and an example of generated σ-algebras is the Borel σ-algebra on the set of real numbers. Consider the collection of closed intervals (denoted with the form [a, b]) on the real numbers. This collection of closed intervals generates the Borel σ-algebra over the real numbers. One can prove that the Borel σ-algebra over the real numbers contains all open intervals, closed intervals, countably infinite unions or intersections of either, and so on. This σ-algebra is, incidentally, the domain of the length measure.
Coda
σ-algebras play an important role in the definition of measure, which plays an important role in the modern theory of integration, Lebesgue integration, which is a cornerstone of the major branch of mathematics, analysis. The use of σ-algebras allows the mathematician to restrict his attention to a smaller, and usually more useful, collection of subsets of a given set. The measures that take a σ-algebra as domain can, then, hope to ignore some of the subsets that are difficult to use. This selectivity provides power to measure and analysis.
Copyright © 2003 by Jimmy Tseng
Book Resources On sigma-algebraIntroductory Real Analysis by A.N. Kolmogorov and S.V. Fomin
Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland
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