your betting strategy in advance of the next flip. The observed information at that point can be described in terms of the 2n possibilities for the first flips. Formally, since you need to use subsets of Ω, this is codified as the σ-algebra Observe that then where is the smallest σ-algebra containing all the others. Definition and properties Definition Let be some set, and let represent its power set. Then a subset is called a σ-algebra if it satisfies the following three properties: is in and is considered to be the universal set in the following context. is closed under complementation: If is in then so is its complement, is closed under countable unions: If are in then so is 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) is in and (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 The largest possible σ-algebra on is Elements of the σ-algebra are called measurable sets. An ordered pair where is a set and is a σ-algebra over 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 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). Dynkin's π-λ theorem This theorem (or the related monotone class theorem) is an essential tool for proving many results about properties of specific σ-algebras. It capitalizes on the nature of two simpler classes of sets, namely the following. A π-system is a collection of subsets of that is closed under finitely many intersections, and A Dynkin system (or λ-system) is a collection of subsets of that contains and is closed under complement and under countable unions of disjoint subsets. Dynkin's π-λ theorem says, if is a π-system and is a Dynkin system that contains then the σ-algebra generated by is contained in Since certain π-systems are relatively simple classes, it may not be hard to verify that all sets in enjoy