# Borel Sets

Definition (Borel Sigma-Algebra, Borel Set)

The

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

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

Definition (Borel Function)

If $(X,A)$ is a measure space and $Y$ is a topological space, then a function $f:X→Y$ is called

measurable, or aBorel function, if it is measurable with respect to $A$ and the Borel σ-algebra on $Y$.

Definition (Borel Measure)

A

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