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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
commit | 28407333ffceca9b99fae721c30e8ae146a863da (patch) | |
tree | 67fa2b79d5c48b50d4e394858af79c88c1447e51 /pages/measure-and-integration/measure-theory/borels-sets.md | |
parent | 777f9d3fd8caf56e6bc6999a4b05379307d0733f (diff) | |
download | site-28407333ffceca9b99fae721c30e8ae146a863da.tar.zst |
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diff --git a/pages/measure-and-integration/measure-theory/borels-sets.md b/pages/measure-and-integration/measure-theory/borels-sets.md new file mode 100644 index 0000000..737a7c8 --- /dev/null +++ b/pages/measure-and-integration/measure-theory/borels-sets.md @@ -0,0 +1,33 @@ +--- +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 %} |