Editing Σ-algebra (section)

===Sub σ-algebras===
In much of probability, especially when [[conditional expectation]] is involved, one is concerned with sets that represent only part of all the possible information that can be observed. This partial information can be characterized with a smaller σ-algebra which is a subset of the principal σ-algebra; it consists of the collection of subsets relevant only to and determined only by the partial information. Formally, if <math>\Sigma, \Sigma'</math> are σ-algebras on <math>X</math>, then <math>\Sigma'</math> is a sub σ-algebra of <math>\Sigma</math> if <math>\Sigma' \subseteq \Sigma</math>. 

The [[Bernoulli process]] provides a simple example. This consists of a sequence of random coin flips, coming up Heads (<math>H</math>) or Tails (<math>T</math>), of unbounded length. The [[sample space]] Ω consists of all possible infinite sequences of <math>H</math> or <math>T:</math>
<math display=block>\Omega = \{H, T\}^\infty = \{(x_1, x_2, x_3, \dots) : x_i \in \{H, T\}, i \geq 1\}.</math>

The full sigma algebra can be generated from an ascending sequence of subalgebras, by considering the information that might be obtained after observing some or all of the first <math>n</math> coin flips. This sequence of subalgebras is given by
<math display=block>
\mathcal{G}_n = \{A \times \{ \Omega \} : A \subseteq \{H, T\}^n\}
</math>
Each of these is finer than the last, and so can be ordered as a [[Filtration (probability theory)|filtration]]

<math display=block>\mathcal{G}_0 \subseteq \mathcal{G}_1 \subseteq \mathcal{G}_2 \subseteq \cdots \subseteq \mathcal{G}_\infty</math>

The first subalgebra <math>\mathcal{G}_0 = \{\varnothing, \Omega\}</math> is the trivial algebra: it has only two elements in it, the empty set and the total space. The second subalgebra <math>\mathcal{G}_1</math> has four elements: the two in <math>\mathcal{G}_0</math> plus two more: sequences that start with <math>H</math> and sequences that start with <math>T</math>.  Each subalgebra is finer than the last. The <math>n</math>'th subalgebra contains <math>2^{n+1}</math> elements: it divides the total space <math>\Omega</math> into all of the possible sequences that might have been observed after <math>n</math> flips, including the possible non-observation of some of the flips.

The limiting algebra <math>\mathcal{G}_\infty</math> is the smallest σ-algebra containing all the others. It is the algebra generated by the [[product topology]] or [[weak topology]] on the product space <math>\{H,T\}^\infty.</math>