pages/measure-and-integration/measure-theory/borels-sets.md


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---
title: Borel Sets
parent: Measure Theory
grand_parent: Measure and Integration
nav_order: 3
---

# {{ page.title }}

{% 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*.
{% enddefinition %}

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 $Y$.
{% enddefinition %}

{% definition Borel Measure %}
A *Borel measure* on a topological space $X$
is any measure on the Borel σ-algebra of $X$.
{% enddefinition %}