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Definition[edit | edit source]

Let X be some set, and let 2X represent its power set. Then a subset is called a σ-algebra if it satisfies the following three properties:[1][2].

  1. X is in , and X is considered to be the universal set in the following context.
  2. Σ is closed under complementation: If A is in Σ, then so is its complement, X \ A.
  3. is closed under countable unions: If A1, A2, A3, ... are in , then so is

Lemmas[edit | edit source]

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 is in , since by

  • (1) X is in
  • (2) asserts that its complement, the empty set, is also in . Moreover, since satisfies condition
  • (3) as well, it follows that is the smallest possible σ-algebra on X.
  • The largest possible σ-algebra is the power set on X, which contains is elements, if X is finite and contains n elementss.

Elements of the -algebra are called measurable sets. An ordered pair , where X is a set and is a -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 -algebra to [0, ∞].

A -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).

Learning Task[edit | edit source]

  • Explore the measurement problem and explain why a -algebra is necessary for the definition of probability distributions!
  • Explain why a discrete probability distribution can use the

References[edit | edit source]

  1. Rudin, Walter (1987). Real & Complex Analysis. McGraw-Hill. ISBN 0-07-054234-1. 
  2. Sigma-algebra. (2017, September 11). In Wikipedia, The Free Encyclopedia. Retrieved 10:03, December 31, 2017, from