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\begin{document}

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\section*{localizable measure}

\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{localizable_measures}{}\section*{{Localizable measures}}\label{localizable_measures}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{other_localizable_structures}{Other localizable structures}\dotfill \pageref*{other_localizable_structures} \linebreak
\noindent\hyperlink{in_weak_foundations}{In weak foundations}\dotfill \pageref*{in_weak_foundations} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Many of the important theorems of [[measure theory]] fail to hold in full generality. (See below under Theorems for which theorems we're talking about.) Often these theorems are stated for $\sigma$-[[sigma-finite measures|finite measures]], but they do hold a bit more generally than that. In fact, they hold for \emph{localizable} measures, and this fact \emph{characterizes} the localizable measures.

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

The following definition is in elementary terms; but we will see that there are many other characterizations.

Let $\mu$ be a [[positive measure]] on an [[abstract set]] $X$. (That is, certain [[subsets]] of $X$, forming a $\sigma$-[[sigma-algebra|algebra]], are [[measurable set|measurable]] by $\mu$, and $\mu$ maps these sets to the space $[0,\infty]$ of [[lower real numbers]] in a [[monotone function|monotone]] and [[countably additive measure|countably additive]] way.)

Given two measurable subsets $E$ and $F$, $E$ \textbf{essentially contains} $F$ if the set

\begin{displaymath}
\{ x\colon X \;|\; x \in F \;\Rightarrow\; x \in E \}
\end{displaymath}
is [[full set|full]]; or equivalently (using [[excluded middle]]) if $F \setminus E$ is [[null set|null]]. (This is a [[preorder]] on the measurable sets.)

Then $\mu$ is \textbf{localizable} if the following conditions both apply:

\begin{itemize}%
\item Semifiniteness: Every measurable set $E$ with positive measure essentially contains a measurable set with finite positive measure. (We may strengthen `essentially contains' to `contains' in this clause.)


\item Essential suprema: Given any collection $\mathcal{C}$ of measurable sets, there is a measurable set $E$ such that:

\begin{itemize}%
\item $E$ essentially contains each element of $\mathcal{C}$; and
\item Given any measurable set $F$ that essentially contains every element of $\mathcal{C}$, $F$ essentially contains $E$.

\end{itemize}
(This set $E$ is essentially unique, in that it essentially contains and is essentially contained in any other set with the same property; we call $E$ [[the]] \textbf{essential union} $\ess \bigcup \mathcal{C}$ of $\mathcal{C}$.)



\end{itemize}
We generalize to [[measures]] taking place in some space other than $[0,\infty]$: a $\mu$ is \textbf{localizable} if ${|\mu|}$ is, as long as $\mu$ has an [[absolute value]] (or [[total variation]]) ${|\mu|}$ that takes values in $[0,\infty]$.

Of course, a set equipped with a localizable measure is a \textbf{localizable measure space}.

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

Every $\sigma$-[[sigma-finite measure|finite]] measure is localizable. (Since this includes so many examples, the theorems below are often stated for $\sigma$-finite measures.)

Any [[counting measure]] is localizable (but $\sigma$-finite only on a [[countable set]]).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

The following [[sheaf]] condition is fundamental: Given any [[cover]] $\mathcal{U}$ of $X$ (a localizable measure space) by [[measurable sets]] and a $\mathcal{U}$-indexed [[family]] $f$ of [[partial function|partial]] [[measurable functions]] (with $\dom f_A = A$), if always $f_A = f_B$ almost everywhere on $A \cap B$ (meaning that there is a [[full subset]] $E$ of $X$ such that $f_A = f_B$ on $A \cap B \cap E$), then there exists a (necessarily unique up to [[almost equality]]) measurable (and total) function $\ess \bigcup f$ such that always $\ess \bigcup f = f_A$ almost everywhere on $A$. (This is Theorem 213N in \hyperlink{Fremlin}{Fremlin}. I don't know if it characterizes localizable measures.)

Slightly more generally, start with \emph{any} family of partial measurable functions on $X$ and treat it as a cover of (any representative of) the essential union of its domains.

\ldots{} other results such as the [[Radon–Nikodym theorem]] \ldots{}

These ideas are elaborated at [[measurable space]] [[Boolean toposes]] and [[Boolean valued model]]s and [[forcing]].

\hypertarget{other_localizable_structures}{}\subsection*{{Other localizable structures}}\label{other_localizable_structures}

Whether a semifinite measure $\mu$ is localizable depends only on which measurable sets are [[full set|full]] (or [[null set|null]]). Sometimes it is convenient to equip a [[measurable space]] with a $\delta$-[[delta-filter|filter]] of full sets (or a $\sigma$-[[sigma-ideal|ideal]] of null sets), without equipping it with the measure of any other set. A \textbf{localizable measurable space} is a measurable space so equipped, such that every collection of measurable sets has an essential union. (It is a theorem, I believe, that every localizable measurable space is capable of supporting a semifinite, hence localizable, measure with the same full/null sets.)

Knowing the full/null sets is also sufficient to define [[almost equality]] of [[measurable functions]] between measurable spaces; this gives us a [[category]] $Loc Meas$ of localizable measurable spaces. This category is [[dual category|dual]] to the category of [[commutative ring|commutative]] [[von Neumann algebras]]; hence an arbitrary von Neumann algebra may be viewed as a \emph{noncommutative} localizable measurable space (in the sense of [[noncommutative geometry]]).

If $\mathcal{M}$ is the $\sigma$-[[sigma-algebra|algebra]] of measurable sets and $\mathcal{F}$ is the $\delta$-filter of full sets (or $\mathcal{N}$ is the $\sigma$-ideal of null sets), then we may form the [[quotient ring|quotient]] [[boolean algebra]] $\mathcal{M}/\mathcal{F}$ (or $\mathcal{M}/\mathcal{N}$) by identifying each full set with the [[improper subset|entire space]] (or identifying each null set with the [[empty subset|empty set]]). Then a measurable space is localizable iff this quotient is [[complete boolean algebra|complete]]; this is the real point of the notion of localizability.

This boolean algebra is all the structure needed to specify a measure on the original space (at least one that is [[absolutely continuous measure|absolutely continuous]] in that it has at least the requisite full/null sets). We can now abstract away the [[underlying set]] and simply look at the complete boolean algebra. However, it is \emph{not} true that every complete boolean algebra is capable of arising in this way from a measurable space. Those which do so arise may be called \textbf{[[measurable locales]]}, since a complete boolean algebra is a [[frame]] and measurable functions (up to almost equality) between the measurable spaces correspond to [[continuous maps]] between the frames, thought of as [[locales]]. (That is, $Loc Meas$ is [[equivalence of categories|equivalent]] to $Meas Loc$, a sort of pun.)

Without requiring localizability, the boolean algebra $\mathcal{M}/\mathcal{F}$ (or $\mathcal{M}/\mathcal{N}$) is called a [[measurable algebra]]; equipped with a measure, we have a [[measure algebra]]. Thus a measurable locale is the same thing as a \textbf{localizable measurable algebra}, and one may also speak of \textbf{localizable measure algebras} (with the category of these equivalent to the category localizable measure spaces, so long as morphisms in the latter are again only defined up to almost equality).

\hypertarget{in_weak_foundations}{}\subsection*{{In weak foundations}}\label{in_weak_foundations}

The [[axiom of choice]] is indispensable for the development above, as stated. (The reason is that one constantly makes choices among essentially equivalent measurable sets, or among almost equal measurable functions.) However, the [[category]] of localizable measurable spaces (and [[measurable functions]] up to [[almost equality]]) is (assuming choice) [[equivalence of categories|equivalent to]] the category of [[measurable locales]], which may prove to be more tractable without choice, even in [[constructive mathematics]]. That said, nobody has worked out a constructive development of this yet. (In particular, identifying which boolean algebras we want is more difficult; perhaps surprisingly, they are still boolean, but they are no longer necessarily complete!)

\hypertarget{references}{}\subsection*{{References}}\label{references}

This giant treatise on all of [[measure theory]] is free (in both senses) online:

\begin{itemize}%
\item D.H. Fremlin, \emph{Measure Theory}, \href{https://www1.essex.ac.uk/maths/people/fremlin/mt.htm}{web}

\end{itemize}
[[!redirects localizable measure]] [[!redirects localizable measures]] [[!redirects localisable measure]] [[!redirects localizable measures]]

[[!redirects localizable measure space]] [[!redirects localizable measure spaces]] [[!redirects localisable measure space]] [[!redirects localizable measure spaces]]

[[!redirects localizable measurable space]] [[!redirects localizable measurable spaces]] [[!redirects localisable measurable space]] [[!redirects localisable measurable spaces]]

[[!redirects localizable measure algebra]] [[!redirects localizable measure algebras]] [[!redirects localisable measure algebra]] [[!redirects localisable measure algebras]]



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