Sigmaalgebras

Sigmaalgebras, or sigma-algebras, are a fundamental concept in measure theory and probability. A sigma-algebra on a set X is a nonempty collection F of subsets of X such that X ∈ F, closed under complementation: if A ∈ F then X \ A ∈ F, and closed under countable unions: if A1, A2, ... ∈ F, then ⋃_{n=1}^∞ A_n ∈ F. From these, F is also closed under countable intersections, via De Morgan's laws. The elements of F are called measurable sets or events.

For any collection G of subsets of X, the sigma-algebra generated by G, denoted σ(G), is the smallest sigma-algebra containing G. This construction provides a convenient way to specify a sigma-algebra without listing all its sets.

Common examples: the trivial sigma-algebra {∅, X}; the discrete or power-set sigma-algebra P(X); on the real line R, the Borel sigma-algebra generated by the open sets, which is the smallest sigma-algebra containing all open subsets of R.

A measurable space is a pair (X, F) where F is a sigma-algebra on X. A function f: X → Y between measurable spaces is measurable if the preimage of every measurable set is measurable; i.e., for all B ∈ G (the sigma-algebra on Y), f^{-1}(B) ∈ F. In probability, one typically takes X as a sample space with a sigma-algebra of events and defines a probability measure on that sigma-algebra.

Sigma-algebras provide the natural domain for measures and integration, and the generation of sigma-algebras by collections of sets underlies many constructions, such as defining Borel sets, push-forwards of measures, and formalizing convergence concepts in probability theory.