Delta-algebra
by Jimmy Tseng
δ-algebra, a mathematical concept, is a collection of subsets of a given set. An alternate name for this concept is delta-algebra since δ is the Greek letter called delta in English. δ-algebra is defined in analysis, one of the major branches of modern mathematics. It is equivalent to the much more popular mathematical concept called σ-algebra, σ-field, sigma-algebra, or sigma-field (these latter four terms are interchangeable). A few mathematicians prefer using δ-algebras instead of σ-algebras to define measure, a key concept in analysis.
The Definition of δ-algebra
Let X be a non-empty set. A δ-algebra over a set X is a non-empty collection, A, of subsets of X that is closed under complements and countably infinite intersections.
The terms δ-algebra and delta-algebra are interchangeable. The term “closed under complements” means that for any element of A, its complement is in A. The term “closed under countably infinite intersections” means that for any countably infinite sequence of elements of A, its intersection is in A.
The Equivalence of δ-algebra and σ-algebra
Let X be a non-empty set. Let A be a δ-algebra over X. If {En} is a countably infinite sequence of elements of A, then {En’}, where apostrophe denotes set complement in X, is a countably infinite sequence of elements of A since A is closed under complements. Hence, E1’ ∩ E2’ ∩ … ∩ En’ … is in A since A is closed under countably infinite intersections. By de Morgan’s law, E1 U E2 U … U En … is also in A. Hence A is closed under countably infinite unions too. Therefore, A is also a σ-algebra. A similar argument will show that a σ-algebra is also a δ-algebra. These two results imply that δ-algebra and σ-algebra are equivalent.
Copyright © 2003 by Jimmy Tseng
Book Resources On Delta-algebraIntroductory Real Analysis by A.N. Kolmogorov and S.V. Fomin
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