Sigma Field

A Sigma Field is a nonempty finite subset of a set's power set.

• AKA: Σ, σ-Algebra, Sigma-Algebra, Ƒ.
• Context:
  • It can be a part of a Field of Sets.
  • It can be the PowerSet of the Random Experiment's Sample Space (Ƒ = 2^Ω)
  • It can be interpreted as the collection of events which can be assigned probabilities.
• Example(s):
  • Σ({a,b,c,d}, { Ø, {a, b}, {a, c, d} }).
  • Σ({a,b,c,d}, { {a, b}, {b, c}, {c, d}, {b, c, d}, {a, b, c, d} }).
• See: Sample Space, Probability Space, Set Complement, Algebra of Sets, Completeness (Order Theory), Field of Sets.

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References

2015

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Sigma-algebra Retrieved:2015-6-4.
  • In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that includes the empty subset, is closed under complement, and is closed under countable unions and countable intersections. The pair (X, Σ) is called a measurable space.

(...) The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for which a given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as the foundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events which can be assigned probabilities. Also, in probability, σ-algebras are pivotal in the definition of conditional expectation.

(...) 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 a measurable function if the preimage of every measurable set is measurable. The collection of measurable spaces forms a category, with the measurable functions as morphisms. Measures are defined as certain types of functions from a σ-algebra to [0, ∞].

A σ-algebra is both a π-system and a Dynkin system (λ-system). The converse is true as well, by Dynkin's theorem (below)

(...) One of the most fundamental uses of the π-λ theorem is to show equivalence of separately defined measures or integrals. For example, it is used to equate a probability for a random variable X with the Lebesgue-Stieltjes integral typically associated with computing the probability:

[math]\displaystyle{ \mathbb{P}(X\in A)=\int_A \,F(dx) }[/math] for all A in the Borel σ-algebra on R,

where F(x) is the cumulative distribution function for X, defined on R, while [math]\displaystyle{ \mathbb{P} }[/math] is a probability measure, defined on a σ-algebra Σ of subsets of some sample space Ω.

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