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\section*{sigma-algebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{measure_and_probability_theory}{}\paragraph*{{Measure and probability theory}}\label{measure_and_probability_theory}

[[!include measure theory - contents]]

\hypertarget{algebras}{}\section*{{$\sigma$-Algebras}}\label{algebras}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{short_version}{Short version}\dotfill \pageref*{short_version} \linebreak
\noindent\hyperlink{long_version}{Long version}\dotfill \pageref*{long_version} \linebreak
\noindent\hyperlink{measurable_sets}{Measurable sets}\dotfill \pageref*{measurable_sets} \linebreak
\noindent\hyperlink{generating_algebras}{Generating $\sigma$-algebras}\dotfill \pageref*{generating_algebras} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{alternatives}{Alternatives}\dotfill \pageref*{alternatives} \linebreak


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

$\sigma$-algebras and their variants are collections of [[subsets]] important in classical [[measure theory]] and [[probability theory]].

Although $\sigma$-algebras are often introduced as a mere technicality in the definition of [[measurable space]] (to specify the [[measurable subsets]]), even once one has a fixed measurable space $X$, it is often useful to consider additional (typically coarser) $\sigma$-algebras of measurable subsets of $X$.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

We assume the law of [[excluded middle]] throughout; see [[Cheng measurable space]] for the constructive theory, and compare also [[measurable locale]].

\hypertarget{short_version}{}\subsubsection*{{Short version}}\label{short_version}

Given a [[set]] $X$, a \textbf{$\sigma$-algebra} is a collection of [[subsets]] of $X$ that is closed under [[complement|complementation]] and under [[unions]] and [[intersections]] of [[countable set|countable]] [[family of subsets|families]].

Notice that the [[power set]] $\mathcal{P} X$ of $X$ is a [[Boolean algebra]] under the operations of complementation and of union and intersection of [[finite set|finite]] families. Actually, it is a [[complete lattice|complete]] Boolean algebra, since we can also take unions and intersections of all families. A $\sigma$-algebra is an intermediate notion, since (in addition to being closed under complementation) we require that it be closed under unions and intersections of \emph{countable} families.

\hypertarget{long_version}{}\subsubsection*{{Long version}}\label{long_version}

Given a [[set]] $X$ and a collection $\mathcal{M}$ of [[subsets]] $S \subseteq X$, there are really several kinds of collections that $\mathcal{M}$ could be:

\begin{itemize}%
\item A \textbf{ring} on $X$ is a collection $\mathcal{M}$ which is closed under [[relative complement]]ation and under [[unions]] of [[finite set|finitary]] [[family of subsets|families]]. That is:

\begin{enumerate}%
\item The [[empty set]] $\empty$ is in $\mathcal{M}$.
\item If $S$ and $T$ are in $\mathcal{M}$, then so is their [[union]] $S \cup T$.
\item If $S$ and $T$ are in $\mathcal{M}$, then so is their [[relative complement]] $T \setminus S$.

\end{enumerate}
It follows that $\mathcal{M}$ is closed under intersections of [[inhabited set|inhabited]] finite families and under [[symmetric difference]] of finite families: * If $S$ and $T$ are in $\mathcal{M}$, then so is their intersection $S \cap T = T \setminus (T \setminus S)$. * If $S$ and $T$ are in $\mathcal{M}$, then so is their symmetric difference $S \uplus T = (T \setminus S) \cup (S \setminus T)$.

We can actually use the latter as an alternative to (2), since $S \cup T = (S \uplus T) \uplus (S \cap T)$. Or we can use the pair as an alternative to (2,3), since $T \setminus S = (S \cap T) \uplus T$. For that matter, we can weaken (1) to simply say that \emph{some} set $S$ is in $\mathcal{M}$; then $\empty = S \setminus S$.

While the union and symmetric difference of an [[empty family]] (both the empty set) belong to $\mathcal{M}$, the intersection of an empty family (which is $S$ itself) might not. The term `ring' dates from the days when a [[ring]] in algebra was not assumed to be unital; so a ring on $X$ is simply a subring (in this sense) of the [[Boolean ring]] $\mathcal{P} X$.


\item A \textbf{$\delta$-ring} on $X$ is a ring (as above) $\mathcal{M}$ which is closed under [[intersections]] of countably infinite families. That is:

\begin{enumerate}%
\item The [[empty set]] $\empty$ is in $\mathcal{M}$.
\item If $S$ and $T$ are in $\mathcal{M}$, then so is their [[union]] $S \cup T$.
\item If $S$ and $T$ are in $\mathcal{M}$, then so is their [[relative complement]] $T \setminus S$.
\item If $S_1, S_2, S_3, \ldots$ are in $\mathcal{M}$, then so is their [[intersection]] $\bigcap_i S_i$.

\end{enumerate}
Of course, every $\delta$-ring is a ring, but not conversely. Actually, if you want to define the concept of $\delta$-ring directly, it's quicker if you use the symmetric difference; then (2,3) follow by the reasoning above and the [[idempotent|idempotence]] of intersection (so that $S \cap T = S \cap T \cap T \cap T \cap \cdots$).

The symbol `$\delta$' here is from German `Durchschnitt', meaning intersection; it may be used in many contexts to refer to intersections of countable families.


\item A \textbf{$\sigma$-ring} on $X$ is a ring (as above) $\mathcal{M}$ which is closed under unions of [[countably infinite set|countably infinite]] families. That is:

\begin{enumerate}%
\item The [[empty set]] $\empty$ is in $\mathcal{M}$.
\item If $S$ and $T$ are in $\mathcal{M}$, then so is their [[union]] $S \cup T$.
\item If $S$ and $T$ are in $\mathcal{M}$, then so is their [[relative complement]] $T \setminus S$.
\item If $S_1, S_2, S_3, \ldots$ are in $\mathcal{M}$, then so is their [[union]] $\bigcup_i S_i$.

\end{enumerate}
Now (2) is simply redundant; $S \cup T = S \cup T \cup T \cup T \cup \cdots$. A $\sigma$-ring is obviously a ring, but in fact it is also a $\delta$-ring; $\bigcap_i S_i = (\bigcup_i S_i) \setminus \bigcup_j (\bigcup_i S_i \setminus S_j)$.

The symbol `$\sigma$' here is from German `Summe', meaning union; it may be used in many contexts to refer to unions of countable families.


\item An \textbf{algebra} or \textbf{field} on $X$ is a ring (as above) $\mathcal{M}$ to which $X$ itself belongs. That is:

\begin{enumerate}%
\item The [[empty set]] $\empty$ is in $\mathcal{M}$.
\item If $S$ and $T$ are in $\mathcal{M}$, then so is their [[union]] $S \cup T$.
\item If $S$ and $T$ are in $\mathcal{M}$, then so is their [[relative complement]] $T \setminus S$.
\item The [[improper subset]] $X$ is in $\mathcal{M}$.

\end{enumerate}
Actually, (2) is now redundant again; $S \cup T = X \setminus ((X \setminus T) \setminus S)$. But perhaps more importantly, $\mathcal{M}$ is closed under \emph{absolute} [[complement|complementation]] (that is, complementation relative to the entire ambient set $X$); that is:

\begin{itemize}%
\item If $S$ is in $\mathcal{M}$, then so is its [[complement]] $\neg{S}$.

\end{itemize}
In light of this, the most common definition of algebra is probably to use this fact together with (1,2); then (3) follows because $T \setminus S = \neg(S \cup \neg{T})$ and (4) follows because $X = \neg\empty$. On the other hand, one could equally well use intersection instead of union; absolute complements allow the full use of [[de Morgan duality]].

The term `field' here is even more archaic than the term `ring' above; indeed the only field in this sense which is a [[field]] (in the usual sense) under symmetric difference and intersection is the field $\{\empty, X\}$ (for an [[inhabited set]] $X$).


\item Finally, a \textbf{$\sigma$-algebra} or \textbf{$\sigma$-field} on $X$ is a ring $\mathcal{M}$ that is both an algebra (as above) and a $\sigma$-ring (as above). That is:

\begin{enumerate}%
\item The [[empty set]] $\empty$ is in $\mathcal{M}$.
\item If $S$ and $T$ are in $\mathcal{M}$, then so is their [[union]] $S \cup T$.
\item If $S$ and $T$ are in $\mathcal{M}$, then so is their [[relative complement]] $T \setminus S$.
\item The [[improper subset]] $X$ is in $\mathcal{M}$.
\item If $S_1, S_2, S_3, \ldots$ are in $\mathcal{M}$, then so is their [[union]] $\bigcup_i S_i$.

\end{enumerate}
As with $\sigma$-rings, (2) is redundant; as with algebras, it's probably most common to use the absolute complement in place of (3,4). Thus the usual definition of a $\sigma$-algebra states: 1. The [[empty set]] $\empty$ is in $\mathcal{M}$. 2. If $S$ is in $\mathcal{M}$, then so is its [[complement]] $\neg{S}$. 3. If $S_1, S_2, S_3, \ldots$ are in $\mathcal{M}$, then so is their [[union]] $\bigcup_i S_i$.

And again we could again just as easily use [[intersection]] as union, even in the infinitary axiom. That is, a $\delta$-algebra is automatically a $\sigma$-algebra, because $\bigcup_i S_i = \neg\bigcap_i \neg{S_i}$.



\end{itemize}
Any and all of the above notions have been used by various authors in the definition of measurable space; for example, Kolmogorov used algebras (at least at first), and Halmos used $\sigma$-rings. Of course, the finitary notions (ring and algebra) aren't strong enough to describe the interesting features of [[Lebesgue measure]]; they are usually used to study very different examples ([[finitely additive measure]]s). On the other hand, $\delta$ or $\sigma$-rings may be more convenient than $\sigma$-algebras for some purposes; for example, [[topological vector space|vector]]-valued measures on $\delta$-rings make good sense even when the absolute measure of the whole space is infinite.

Note that the collection of measurable sets with finite measure (in a given [[measure space]]) is a $\delta$-ring, while the collection of measurable sets with $\sigma$-finite measure is a $\sigma$-ring.

\hypertarget{measurable_sets}{}\subsubsection*{{Measurable sets}}\label{measurable_sets}

A \textbf{[[measurable space]]} is usually defined to be a [[set]] $X$ with a $\sigma$-algebra $\mathcal{M}$ on $X$; sometimes one of the more general variations above is used.

In any case, an \textbf{$\mathcal{M}$-[[measurable subset]] of $X$}, or just a \textbf{measurable set}, is any [[subset]] of $X$ that belongs to $\mathcal{M}$. If $\mathcal{M}$ is one of the more general variations, then we also want some subsidiary notions: $S$ is \textbf{[[relatively measurable subset|relatively measurable]]} if $S \cap T$ belongs to $\mathcal{M}$ whenever $T$ does, and $S$ is \textbf{$\sigma$-[[sigma-measurable subset|measurable]]} if it is a countable union of elements of $\mathcal{M}$. Notice that every relatively measurable set is measurable iff $S$ is at least an algebra; in any case, the relatively measurable sets form a ($\sigma$)-algebra if $\mathcal{M}$ is at least a ($\delta$)-ring. Similary, every $\sigma$-measurable set is measurable iff $S$ is at least a $\sigma$-ring; in any case, the $\sigma$-measurable sets form a $\sigma$-ring if $\mathcal{M}$ is at least a $\delta$-ring.

\hypertarget{generating_algebras}{}\subsubsection*{{Generating $\sigma$-algebras}}\label{generating_algebras}

As a $\sigma$-algebra is a collection of subsets, we might hope to develop a theory of [[base|bases]] and [[subbase|subbases]] of $\sigma$-algebras, such as is done for [[topological space|topologies]] and [[uniform space|uniformities]]. However, things do not work out as nicely. (It \emph{is} quite easy to generate rings or algebras, but generating $\delta$-rings and $\sigma$-rings is just as tricky as generating $\sigma$-algebras.)

We do get something by [[general abstract nonsense]], of course. It's easy to see that the [[intersection]] of any collection of $\sigma$-algebras is itself a $\sigma$-algebra; that is, we have a [[Moore closure]]. So given any collection $\mathcal{B}$ of sets whatsoever, the intersection of all $\sigma$-algebras containing $\mathcal{B}$ is a $\sigma$-algebra, the $\sigma$-algebra \textbf{generated} by $\mathcal{B}$. (We can similarly define the $\delta$-ring generated by $\mathcal{B}$ and similar concepts for all of the other notions defined above.)

What is missing is a simple description of the $\sigma$-algebra generated by $\mathcal{B}$. (For a mere algebra, this is easy; any $\mathcal{B}$ can be taken as a \emph{[[subbase]]} of an algebra, the symmetric unions of finite families of elements of $\mathcal{B}$ form a \emph{[[base]]} of the algebra, and the intersections of finite families of elements of the base form an algebra. For a ring, the only difference is to use intersections only of inhabited families. But for anything from a $\delta$-ring to a $\sigma$-algebra, nothing this simple will work.)

In fact, the question of how to generate a $\sigma$-algebra is the beginning of an entire field of mathematics, [[descriptive set theory]]. For our purposes, we need this much:

\begin{itemize}%
\item Start with a collection $\Sigma_0$ (our collection $\mathcal{B}$ above), and let $\Pi_0$ be the collection of the complements of the members of $\Sigma_0$.
\item Let $\Sigma_1$ be the collection of unions of countably infinite families of sets in $\Pi_0$, and let $\Pi_1$ be the collection of their complements (the intersections of countably infinite families of sets in $\Sigma_0$); even $\Sigma_1 \cup \Pi_1$ is not in general a $\sigma$-algebra.
\item Continue by [[recursion|recursively]], 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 by [[transfinite induction|transfinite]] recursion, defining $\Sigma_\alpha$ for all countable [[ordinal numbers]] $\alpha$.
\item Let $\Sigma_{\omega_1}$ be the union of the various $\Sigma_\alpha$; \emph{this} is finally a $\sigma$-algebra.

\end{itemize}
So we need an $\aleph_1$ steps, not just $2$.

(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.)

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{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item Of course, the [[power set]] of $X$ is closed under all operations, so it is a $\sigma$-algebra.


\item If $X$ is a [[topological space]], the $\sigma$-algebra generated by the open sets (or equivalently, by the closed sets) in $X$ is the \textbf{Borel $\sigma$-algebra}; its elements are called the \textbf{[[Borel sets]]} of $X$. In particular, the Borel sets of [[real numbers]] are the Borel sets in the [[real line]] with its usual topology.


\item In the application of [[statistical physics]] to [[thermodynamics]], we have both a microcanonical [[phase space]] $P$ (typically something like $\mathbb{R}^N$ for $N$ on the order of [[Avogadro's number]]) describing every last detail of a [[physical system]] and a macrocanonical phase space $p$ (typically $\mathbb{R}^2$ or at least $\mathbb{R}^n$ for $n \lt 10$) describing those features of the system that can be measured practically, with a projection $P \to p$. Then the [[preimage]] under this projection of the Borel $\sigma$-algebra of $p$ is a $\sigma$-algebra on $P$, and the thermodynamic [[entropy]] of the system is (theoretically) its information-theoretic entropy with respect to this $\sigma$-algebra.


\item If a measurable space $(X,\mathcal{M})$ is equipped with a (positive) measure $\mu$, making it into a [[measure space]], then the sets of measure zero form a $\sigma$-[[sigma-ideal|ideal]] of $\mathcal{M}$ (that is an [[ideal]] that is also a sub-$\sigma$-ring). Let a \textbf{[[null set]]} be \emph{any} set (measurable or not) contained in a set of measure zero; then the null sets form a $\sigma$-ideal in the [[power set]] of $X$. Call a set \textbf{$\mu$-measurable} if it is the union of a measurable set and a null set; then the $\mu$-measurable sets form a $\sigma$-algebra called the \textbf{completion} of $\mathcal{M}$ under $\mu$. (Even if $\mathcal{M}$ is only a $\delta$-ring, still the null sets will form a $\sigma$-ring; in any case, we get as completion the same kind of structure as we began with.) Note that we can also do this by starting with any $\sigma$-ideal $\mathcal{N}$ and simply declaring by fiat that these are the null sets, as with a [[localisable measurable space]]; then we speak of the \textbf{completion} of $\mathcal{M}$ with respect to $\mathcal{N}$ (or sometimes with respect to the $\delta$-[[delta-filter|filter]] $\mathcal{F}$ of [[full sets]]).


\item In particular, the \textbf{Lebesgue-measurable} sets in the real line are the elements of the completion of the Borel $\sigma$-algebra under [[Lebesgue measure]].



\end{itemize}
\hypertarget{alternatives}{}\subsection*{{Alternatives}}\label{alternatives}

We are now learning ways to understand measure theory and probability away from the traditional reliance on sets required with $\sigma$-algebras; see [[measurable space]] for a summary of other ways to define this concept. We still need to know what happens to all of the \emph{other} $\sigma$-algebras of measurable sets in a measurable space. One solution may to use [[quotient space|quotient]] measurable spaces in place of sub-$\sigma$-algebras; for example, see explicit quotient in the example of macroscopic entropy above.

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