This post elaborates on the concept of σ-algebra generated by a set, as presented in p. 17 of the book “Probability with martingales” by David Williams.
Let be a class of subsets of a set . It is stated in the book that the σ-algebra generated by is the smallest σ-algebra on such that contains . Moreover, it is mentioned that is the intersection of all σ-algebras on that contain .
Three clarifications will be made. Let be the set of all σ-algebras on that contain .
Firstly, it will be shown that the intersection , which is the intersection of all σ-algebras on that contain , is itself a σ-algebra on that contains . Since for all , it follows that . Moreover, consider a set . So for all , which means for all , so . Let be a countable collection of sets . It follows that for all and all , so for all , which gives . It has thus been confirmed that is a σ-algebra on . As for all , it becomes obvious that , so the σ-algebra on contains .
Consider now those σ-algebras with that satisfy for all with . This is what is meant by “smallest σ-algebra on such that ” in William’s book. Assume that there are two such distinct σ-algebras and , i.e. . Since , it is then deduced that . Similarly, , so , which is a contradiction. Hence, there is a single σ-algebra in that contains and is a subset of any other σ-algebra in that also contains .
It has been shown that . Moreover, satisfies for any . Uniqueness has also been established, so is the only element in that is a subset of all . Put in words, is the only σ-algebra on that contains and is contained in any other σ-algebra on that satisfies .
is called the σ-algebra generated by and is denoted by .