Question 10.T.2: Let D be any class of subsets of Ω. Then there is a smallest...
Let \mathcal{D} be any class of subsets of Ω. Then there is a smallest σ-algebra \mathcal{A}(D) on Ω that contains \mathcal{D}.
Let \mathcal{S} be the family of all σ-algebras containing \mathcal{D}. \mathcal{S} is not empty as \mathcal{P}(Ω) ∈ \mathcal{S}, so we can define \mathcal{A(D)} to be the (non-empty) intersection of all the σ-algebras in \mathcal{S}. Since each σ-algebra in \mathcal{S} contains Ω and is closed under the formation of countable unions and complementation, the same is true of \mathcal{A(D)}. Thus \mathcal{A(D)} is a σ-algebra on Ω and is clearly the smallest such σ-algebra, in the sense that any σ-algebra on Ω that contains \mathcal{D} also contains \mathcal{A(D).}
\mathcal{A(D)} is called the σ-algebra generated by \mathcal{D}. Note, in this context, that the term class and family are used as synonyms of set. A simple modification of the proof yields a smallest ring, a smallest algebra, or a smallest σ-ring containing \mathcal{D}, or generated by \mathcal{D}.