Sigma Algebra Generator I have some confusion regarding the definition of the generator of a sigma algebra. Looking at the wikipedia definition,
'Let F be an arbitrary family of subsets of X. Then there exists a unique smallest σ-algebra which contains every set in F (even though F may or may not itself be a σ-algebra). It is, in fact, the intersection of all σ-algebras containing F. (See intersections of σ-algebras above.) This σ-algebra is denoted σ(F) and is called the σ-algebra generated by F'
So why does this smallest such sigma algebra contain every set in $F$, for example the Borel sigma algebra is generated by $A = \{(a,b): a,b \in \mathbb{R}\}$, why does it then follow that it contains all open intervals from this?
 A: You are misreading the statement. You seem to be reading it as "there exists a unique smallest $\sigma$-algebra" and that this $\sigma$-algebra "contains every set in $F$."
Rather, it should be read as "if we consider all $\sigma$-algebras that satisfy the property 'contains every set in $F$,' then there is a unique smallest $\sigma$-algebra among these $\sigma$-algebras." The proof is to consider the intersection of all these $\sigma$-algebras and recall that the arbitrary intersection of $\sigma$-algebras is itself a $\sigma$-algebra.
A: $\sigma(F)$ contains every set in $F$ by construction. That is, it is designed to contain every set in $F$. You should take the passage's advice and "See intersections of sigma algebras above." I am sure that above the article proves the lemma that any intersection of sigma algebras is again a sigma algebra.
In your particular example, let $\Omega$ be the collection of all sigma algebras $\Sigma$ that satisfy the condition $\Sigma \supset A$. You first need to make sure that $\Omega$ is not empty; then apply the lemma.
Do you now see that we can find every open interval in the family $$\bigcap_{\Sigma \in\Omega} \Sigma \quad ?$$
