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In mathematics, a σ-algebra (or sigma-algebra) (sigma is a Greek letter, upper case Σ, lower case σ) over a set X is a nonempty collection Σ of subsets of X that is closed under complementation and countable unions of its members. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Sigma (upper case Σ, lower case σ; Greek Σιγμα lower case in word-final position ς) is the eighteenth letter of the Greek In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets It is a Boolean algebra, completed to include countably infinite operations. In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. In the mathematical area of Order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered The pair <X, Σ> is also a field of sets, sometimes called a σ-field or a measurable space. In Mathematics a field of sets is a pair \langle X \mathcal{F} \rangle where X is a set and \mathcal{F} is an algebra

Thus, if X={a, b, c, d}, one sigma algebra could be Σ = { ∅, {a, b}, {c, d}, {a, b, c, d} }.

The main use of σ-algebras is in the definition of measures on X. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with The concept is important in mathematical analysis and probability theory. Analysis has its beginnings in the rigorous formulation of Calculus. Probability theory is the branch of Mathematics concerned with analysis of random phenomena

Contents

Definition and properties

Formally, a subset Σ of the power set of a set X is a σ-algebra if and only if it has the following properties:

  1. Σ is nonempty
  2. If E is in Σ then so is the complement (X \ E) of E. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation
  3. The union of countably many sets in Σ is also in Σ.

In other words, a family Σ of subsets of X is a σ-algebra if:

  1. Σ contains X (or, Σ contains the empty set)
  2. Σ is closed under complements
  3. Σ is closed under countable unions. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members

From these axioms, it follows that X and the empty set are in Σ, and that the σ-algebra is also closed under countable intersections (via De Morgan's laws). In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members In Logic, De Morgan's laws or De Morgan's theorem are rules in Formal logic relating pairs of dual Logical operators in a systematic manner expressed

A measure on X is a function which assigns a real number to subsets of X; this can be thought of as making precise a notion of 'size' or 'volume' for sets. One might like to assign such a size to every subset of X, but the axiom of choice implies that when the size under consideration is standard length for subsets of the real line, then there exist sets known as Vitali sets for which no size exists. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. In Mathematics, a Vitali set is an elementary example of a set of Real numbers that is not Lebesgue measurable. For this reason, one considers instead a smaller collection of privileged subsets of X whose measure is defined; these sets constitute the σ-algebra.

Elements of the σ-algebra are called measurable sets. An ordered pair (X, Σ), where X is a set and Σ is a σ-algebra over X, is called a measurable space. A function between two measurable spaces is called measurable if the preimage of every measurable set is measurable. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage The collection of measurable spaces forms a category with the measurable functions as morphisms. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, measurable functions are Well-behaved functions between measurable spaces. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and Measures are defined as certain types of functions from a σ-algebra to [0,∞]. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with

σ-algebras are sometimes denoted using capital letters from a Fraktur typeface. The German word Fraktur () refers to a specific sub-group of Blackletter Typefaces The word derives from the past participle fractus (“broken” Thus, \mathfrak{F} may be used to denote (X,Σ). Another common convention is to use calligraphic capital letters in place of Σ, thus (X,\mathcal{A})\,\; is often used in place of (X,Σ). Calligraphy (from Greek kallos "beauty" + graphẽ "writing" is the art of writing (Mediavilla 1996 17 This is handy to avoid situations where Σ might be confused for the summation operator.

Generated σ-algebra

If U is an arbitrary family of subsets of X then we can form a special σ-algebra from U, called the σ-algebra generated by U. We denote it by σ(U) and define it as follows. First note that there is a σ-algebra over X that contains U, namely the power set of X. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) Let Φ be the family of all σ-algebras over X that contain U (that is, a σ-algebra Σ over X is in Φ if and only if U is a subset of Σ. ) Then we define σ(U) to be the intersection of all σ-algebras in Φ. σ(U) is then the smallest σ-algebra over X that contains U. For a simple example, consider the set X={1,2,3}. Then the σ-algebra generated by the subset {1} is σ({1}) = { ∅, {1}, {2,3}, X}. By an abuse of notation, when the collection of subsets contains only one member, call it A, one may write σ(A) instead of σ({A}).

In general, there is no explicit description of the σ-algebra generated by a given collection. This is in contrast with a similar construction in topology: the smallest topology generated by a family of subsets can be explicitly described. In Highway engineering, subbase is a layer between Subgrade and the Base course. For example, there can be sets in the generated algebra constructed from the generator system only in transfinite number of steps. However, it is a theorem that a generated σ-algebra is the (possibly transfinite) union of all σ-algebras generated by the countable sub-systems of the generator system.

Examples

Let X be any set, then the following are σ-algebras over X:

Examples for generated algebras

An important example is the Borel algebra over any topological space: the σ-algebra generated by the open sets (or, equivalently, by the closed sets). In Mathematics, the Borel algebra (or Borel &sigma-algebra) on a Topological space X is a &sigma-algebra of Subsets of Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Topology and related branches of Mathematics, a closed set is a set whose complement is open. Note that this σ-algebra is not, in general, the whole power set. For a non-trivial example, see the Vitali set. In Mathematics, a Vitali set is an elementary example of a set of Real numbers that is not Lebesgue measurable.

On the Euclidean space Rn, another σ-algebra is of importance: that of all Lebesgue measurable sets. In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to This σ-algebra contains more sets than the Borel σ-algebra on Rn and is preferred in integration theory, as it gives a complete measure space. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Mathematics, a complete measure (or more precisely a complete measure space) is a measure space in which every Subset of every Null

See also

External links

In Mathematics a field of sets is a pair \langle X \mathcal{F} \rangle where X is a set and \mathcal{F} is an algebra In Mathematics, measurable functions are Well-behaved functions between measurable spaces. In Probability theory, the sample space or universal sample space, often denoted S, Ω or U (for "universe" of an Experiment In Mathematics, &sigma-algebras are usually studied in the context of Measure theory. In Mathematics, a nonempty collection of sets \mathcal{R} is called a σ-ring (pronounced sigma-ring) if it is closed under countable In Mathematics, additivity and sigma additivity of a function defined on Subsets of a given set are abstractions of the intuitive properties
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