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+---
+title: Borel Sets
+parent: Measure Theory
+grand_parent: Measure and Integration
+nav_order: 2
+---
+
+# {{ 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 %}