What is a countably generated $\sigma$-algebra? Can't find a definition online What is a countably generated $\sigma$-algebra? Is Borel $\sigma$-algebra countably generated?
 A: A countably generated $\sigma$-algebra is a $\sigma$-algebra $\mathcal M$ of subsets of a set $X$ such that there exists a countable family $\mathcal E=\{E_n:n\in\mathbb N\}$ such that $\mathcal M=\sigma(\mathcal E)$, where $\sigma(\mathcal E)$ denotes the $\sigma$-algebra generated by $\mathcal E$.  
The Borel $\sigma$-algebra won't always be countably generated.  The Borel $\sigma$-algebra will be countably generated if the topological space is second countable. For example, the Borel $\sigma$-algebra of  a separable metric space is countably generated.
A: Any family of subsets $\mathcal{C}$ of some set $X$ generates a unique $\sigma$-algebra $\langle \mathcal{C} \rangle$ on $X$, namely the intersection of all $\sigma$-algebras that contain $\mathcal{C}$ as a subset. (The powerset of $X$ is one of such, and the intersection of $\sigma$-algebras is again a $\sigma$-algebra).
A given $\sigma$-algebra $\mathcal{A}$ is countably generated iff there exists a countable family $\mathcal{C}$ such that $\langle \mathcal{C}\rangle = \mathcal{A}$.
For a second countable space $X$ we have that the Borel algebra is generated by the countable base on $X$ that must exist. So yes for all separable metrisable spaces, including all Euclidean spaces.
