Let X be some set, and let 2X represent its power set. Then a subset Σ ⊆ 2 X {\displaystyle \Sigma \subseteq 2^{X}} is called a σ-algebra if it satisfies the following three properties:. X is in S {\displaystyle {\mathcal {S}}} , and X is considered to be the universal set in the following context. Σ is closed under complementation: If A is in Σ, then so is its complement, X \ A. S {\displaystyle {\mathcal {S}}} is closed under countable unions: If A1, A2, A3, ... are in S {\displaystyle {\mathcal {S}}} , then so is A = A 1 ∪ A 2 ∪ A 3 ∪ ... = ⋃ i = 1 ∞ A i {\displaystyle A=A_{1}\cup A_{2}\cup A_{3}\cup \ldots =\bigcup _{i=1}^{\infty }A_{i}} From these properties, it follows that the σ-algebra is also closed under countable intersections (by applying De Morgan's laws). It also follows that the empty set ∅ {\displaystyle \emptyset } is in S {\displaystyle {\mathcal {S}}} , since by (1) X is in S {\displaystyle {\mathcal {S}}} (2) asserts that its complement, the empty set, is also in S {\displaystyle {\mathcal {S}}} . Moreover, since { X , ∅ } {\displaystyle \{X,\emptyset \}} satisfies condition (3) as well, it follows that { X , ∅ } {\displaystyle \{X,\emptyset \}} is the smallest possible σ-algebra on X. The largest possible σ-algebra is the power set on X, which contains is 2 n {\displaystyle 2^{n}} elements, if X is finite and contains n elementss. Elements of the σ {\displaystyle \sigma } -algebra are called measurable sets. An ordered pair ( X , S ) {\displaystyle (X,{\mathcal {S}})} , where X is a set and S {\displaystyle {\mathcal {S}}} is a σ {\displaystyle \sigma } -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 {\displaystyle \sigma } -algebra to [0, ∞]. A σ {\displaystyle \sigma } -algebra is both a π-system and a Dynkin system (λ-system). The converse is true as well, by Dynkin's theorem (see Wikipedia:Sigma algebra). Explore the measurement problem and explain why a σ {\displaystyle \sigma } -algebra is necessary for the definition of probability distributions! Explain why a discrete probability distribution can use the Rudin, Walter (1987). Real & Complex Analysis. McGraw-Hill. ISBN 0-07-054234-1. Sigma-algebra. (2017, September 11). Retrieved 10:03, December 31, 2017, from https://en.wikipedia.org/w/index.php?title=Sigma-algebra&oldid=800163136
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