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\begin{document}

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\section*{Borel subset}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{the_borel_hierarchy}{The Borel hierarchy}\dotfill \pageref*{the_borel_hierarchy} \linebreak
\noindent\hyperlink{the_borel_ring_and_ring}{The Borel $\delta$-ring and $\sigma$-ring}\dotfill \pageref*{the_borel_ring_and_ring} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Borel sets are certain [[subsets]] of a [[topological space]]. They form the Borel $\sigma$-[[sigma-algebra|algebra]] of the space, and they play an important role in [[measure theory]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $X$ be a [[topological space]]. Then there is a $\sigma$-[[sigma-algebra|algebra]] $\mathcal{B}$ on $X$ generated by the [[open subsets]] of $X$. Elements of $\mathcal{B}$ are called the \textbf{Borel sets} (or \textbf{Borel subsets}, or \textbf{Borel-measurable sets}, etc) of $X$, and $\mathcal{B}$ itself is called the \textbf{Borel $\sigma$-algebra} on $X$.

\hypertarget{the_borel_hierarchy}{}\subsection*{{The Borel hierarchy}}\label{the_borel_hierarchy}

The preceding abstract definition can be made concrete (and [[predicative mathematics|predicative]], at least over $\omega_1$, although most predicative mathematicians don't accept $\omega_1$) as follows:

\begin{itemize}%
\item Start with the collection $\Sigma_0$ of open sets, and let $\Pi_0$ be the collection of the complements of the members of $\Sigma_0$ (the [[closed sets]]).
\item Let $\Sigma_1$ be the collection of countably infinitary unions of sets in $\Pi_0$ (the \textbf{$F_\sigma$-[[F-sigma subspace|sets]]}), and let $\Pi_1$ be the collection of their complements (the \textbf{$G_\delta$-[[G-delta subspace|sets]]}, the countably infinitary intersections of sets in $\Sigma_0$); even $\Sigma_1 \cup \Pi_1$ is not in general a $\sigma$-algebra.
\item Let $\Sigma_2$ be the collection of countably infinitary unions of sets in $\Pi_1$ (the \textbf{$G_{\delta\sigma}$-sets}), and let $\Pi_2$ be the collection of their complements (the \textbf{$F_{\sigma\delta}$-sets}, the countably infinitary intersections of sets in $\Sigma_1$).
\item Continue, defining $\Sigma_n$ for all [[natural numbers]] $n$.
\item Let $\Sigma_\omega$ be the union of the various $\Sigma_n$; although this is closed under complement, it is still not in general a $\sigma$-algebra.
\item Continue, defining $\Sigma_\alpha$ for all countable [[ordinal numbers]] $\alpha$.
\item Let $\Sigma_{\omega_1}$ be the union of the various $\Sigma_\alpha$; this is finally a $\sigma$-algebra.

\end{itemize}
So we need an [[uncountable set|uncountable]] number of steps, not just two.

This is only the beginning of [[descriptive set theory]]; our $\Sigma_\alpha$ are their $\Sigma^0_\alpha$ ---except that for some reason they start with $\Sigma^0_1$ instead of $\Sigma^0_0$---, and the subject continues to higher values of the superscript.

(To remember the other symbols, you need to know two languages: French and German. The `$F$' comes from French `ferm\'e{}' for `closed', while `$G$' is simply the next letter; the `$\sigma$' and `$\delta$' are from German `Summe' for `union' and `Durchschnitt' for `intersection' and are commonly used for countable such.)

Note that [[countable choice]] is essential here and elsewhere in measure theory, to show that a countable union of a countable union is a countable union. But the full [[axiom of choice]] is not; in fact, much of descriptive set theory (although this is irrelevant to the small portion above) works better with the [[axiom of determinacy]] instead.

\hypertarget{the_borel_ring_and_ring}{}\subsection*{{The Borel $\delta$-ring and $\sigma$-ring}}\label{the_borel_ring_and_ring}

Sometimes one does measure theory with $\sigma$-rings or even $\delta$-rings, which are more general than $\sigma$-algebras. However, even the $\delta$-ring generated by a topology is in fact a $\sigma$-algebra.

Nevertheless, on a [[locally compact Hausdorff space]], we may define the \textbf{Borel $\delta$-ring} and the \textbf{Borel $\sigma$-ring} to be the $\delta$-ring or $\sigma$-ring generated by the [[compact space|compact]] subsets of $X$. When $X$ is [[compact Hausdorff space|compact]] (such as the [[unit interval]]), then these both agree with the Borel $\sigma$-algebra; when $X$ is $\sigma$-compact (a countable union of compact subsets, such as the [[real line]]), then the Borel $\sigma$-ring still agrees with the Borel $\sigma$-algebra.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[projective set]]


\item [[Borel measure]]



\end{itemize}
[[!redirects Borel subset]] [[!redirects Borel subsets]] [[!redirects Borel set]] [[!redirects Borel sets]] [[!redirects Borel-measurable subset]] [[!redirects Borel-measurable subsets]] [[!redirects Borel-measurable set]] [[!redirects Borel-measurable sets]]

[[!redirects Borel sigma-algebra]] [[!redirects Borel sigma-algebras]] [[!redirects Borel ∞-algebra]] [[!redirects Borel ∞-algebras]]

[[!redirects Borel hierarchy]] [[!redirects Borel hierarchies]]

[[!redirects Borel sigma-ring]] [[!redirects Borel sigma-rings]] [[!redirects Borel ∞-ring]] [[!redirects Borel ∞-ring]]

[[!redirects Borel delta-ring]] [[!redirects Borel delta-rings]] [[!redirects Borel ∞-ring]] [[!redirects Borel ∞-ring]]



\end{document}