- X is in , 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.
- is closed under countable unions: If A1, A2, A3, ... are in , then so is
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, ∞].
Learning Task edit
- 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
- Rudin, Walter (1987). Real & Complex Analysis. McGraw-Hill. ISBN 0-07-054234-1.
- Sigma-algebra. (2017, September 11). In Wikipedia, The Free Encyclopedia. Retrieved 10:03, December 31, 2017, from https://en.wikipedia.org/w/index.php?title=Sigma-algebra&oldid=800163136