Borel Sets

Definition (Borel Sigma-Algebra, Borel Set)

The Borel σ-algebra $\mathcal{B}(X)$ on a topological space $X$ is the σ-algebra generated by its open sets. The elements of $\mathcal{B}(X)$ are called Borel(-measurable) sets.

That is, $\mathcal{B}(X)=\sigma (\mathcal{O})$, where $\mathcal{O}$ is the collection of open sets in $X$. It is also true that $\mathcal{B}(X)=\sigma (\mathcal{C})$, where $\mathcal{C}$ is the collection of closed sets in $X$.

Definition (Borel Function)

If $(X,\mathcal{A})$ is a measure space and $Y$ is a topological space, then a function $f:X\to Y$ is called measurable, or a Borel function, if it is measurable with respect to $\mathcal{A}$ and the Borel σ-algebra on $Y$.

Definition (Borel Measure)

A Borel measure on a topological space $X$ is any measure on the Borel σ-algebra of $X$.