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\section*{Cheng space}

\hypertarget{cheng_spaces}{}\section*{{Cheng spaces}}\label{cheng_spaces}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{motivation}{Motivation}\dotfill \pageref*{motivation} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{the_boolean_semialgebra_of_disjoint_pairs}{The Boolean semi-algebra of disjoint pairs}\dotfill \pageref*{the_boolean_semialgebra_of_disjoint_pairs} \linebreak
\noindent\hyperlink{the_semialgebra_of_complemented_pairs}{The $\sigma$-semi-algebra of complemented pairs}\dotfill \pageref*{the_semialgebra_of_complemented_pairs} \linebreak
\noindent\hyperlink{measurable_sets_and_functions}{Measurable sets and functions}\dotfill \pageref*{measurable_sets_and_functions} \linebreak
\noindent\hyperlink{completion}{Completion}\dotfill \pageref*{completion} \linebreak
\noindent\hyperlink{measures_on_cheng_spaces}{Measures on Cheng spaces}\dotfill \pageref*{measures_on_cheng_spaces} \linebreak


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

A Cheng space is a version of a [[measurable space]] developed by [[Henry Cheng]] for [[constructive mathematics]]. Even in [[classical mathematics]], however, Cheng spaces are more general than standard measure spaces. On the other hand, if we equip a measurable space with a $\sigma$-[[sigma-ideal|ideal]] of [[null subsets]] (or a $\delta$-filter of [[full subsets]]), there is no essential difference classically.

Some of the [[abstract nonsense]] below is original research (by [[Toby Bartels]] and [[Todd Trimble]]), but based heavily on Cheng's work as described in [[Handbook of Constructive Analysis|Bishop \& Bridges]].

\hypertarget{motivation}{}\subsection*{{Motivation}}\label{motivation}

From a [[constructive mathematics|constructive]] perspective, there are a couple of related problems with the classical theory of measurable spaces. One is that the notion of $\sigma$-[[sigma-algebra|algebra]] is highly suspicious, because it relies on an operation, [[complement]]ation, that behaves very differently in [[intuitionistic logic]]. The other is that, even you accept the definition of $\sigma$-algebra (after all, the Lebesgue-measurable sets on the real line do still form one), there may be very few measurable functions. (It is possible in constructive mathematics that every function from the [[real line]] to itself, regardless of measurability, is [[continuous function|continuous]].)

Indeed, if we set aside the general theory of measurable spaces and simply do Lebesgue measure ad hoc in a constructive (even predicative) way, we find that instead of measurable [[functions]] we really want measurable [[partial functions]] whose domain of definition is a [[full subset]], the [[almost functions]]. This suggests that if we want to define the concept of measurable function, then we have to know what the full sets are, so we need a measurable space equipped with such a notion. But since this is usually defined relative to a measure, well \emph{after} the structure of the measurable space has been given, we are at an impasse.

There is a way out, due to [[Henry Cheng]], for both of these problems at once. Instead of dealing with individual subsets, we will deal with pairs of [[disjoint subsets]]. The intuition is that we use disjoint pairs $(A,B)$ such that $A \cup B$ is full ---with $(A,\neg{A})$ being the motivating example in the classical theory---, but we let the Cheng measurability structure itself tell us which pairs those are. Once we fix a particular measure, we may find additional pairs whose union is full, somewhat like finding additional measurable sets when taking the completion in the classical theory (although taking the completion is a separate phenomenon here), but that's all right; the important thing is that each pair chosen really is full in any measure used (much as each set in a classical $\sigma$-algebra must actually be measurable by any measure used).

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

Let $X$ be a [[set]]. We'll define the structure of a Cheng space on $X$ in several steps.

\hypertarget{the_boolean_semialgebra_of_disjoint_pairs}{}\subsubsection*{{The Boolean semi-algebra of disjoint pairs}}\label{the_boolean_semialgebra_of_disjoint_pairs}

A \textbf{[[disjoint subsets|disjoint pair]]} in $X$ is a pair $(A,B)$ of [[subsets]] of $X$ such that the [[intersection]] $A \cap B$ is [[empty subset|empty]]. Every set $A$ defines a disjoint pair $[A] = (A,\neg{A})$, but many disjoint pairs are not of this form; the extreme counterexample (unless $X$ is empty) is $(\empty,\empty)$. We order disjoint pairs by the usual order on the first component and the opposite on the second:

\begin{displaymath}
(A,B) \subseteq (C,D) \;\Leftrightarrow\; A \subseteq C \;\wedge\; B \supseteq D .
\end{displaymath}
Similarly, we make the usual operations on sets into operations on disjoint pairs by applying formal [[de Morgan duality]] to the second component:

\begin{itemize}%
\item $\empty = [\empty] = (\empty, X)$;
\item $X = [X] = (X, \empty)$;
\item $(A,B) \cup (C,D) = (A \cup C, B \cap D)$;
\item $(A,B) \cap (C,D) = (A \cap C, B \cup D)$;
\item $\bigcup_i (A_i,B_i) = (\bigcup_i A_i, \bigcap_i B_i)$;
\item $\bigcap_i (A_i,B_i) = (\bigcap_i A_i, \bigcup_i B_i)$.

\end{itemize}
(Note that we do not write $[A]$ as $A$ except when $A$ is given as $\empty$ or $X$, because for example, $[A \cap B] = [A] \cap [B]$, while classically valid, may fail constructively.)

These operations form the disjoint pairs into a [[lattice]]; in fact, it is both a [[complete lattice]] and a [[distributive lattice]], but it is not constructively completely distributive in either direction. (Compare the fact that a [[power set]] is, constructively, completely distributive only in one direction, making it a [[frame]]; here the directions are mixed by the formal duality and so neither works. On the other hand, that the power set is a frame is used to show that the infinitary operations do define disjoint pairs.)

Finally, we define the [[complement]] of $(A,B)$, not using the complements of $A$ and $B$ (which usually are not even disjoint) but instead simply by reversing them:

\begin{displaymath}
\neg(A,B) = (B,A) .
\end{displaymath}
Then an actual [[de Morgan duality]] holds for these operations:

\begin{itemize}%
\item $\neg\bigcup_i (A_i,B_i) = \bigcap_i \neg(A_i,B_i)$;
\item $\neg\bigcap_i (A_i,B_i) = \bigcup_i \neg(A_i,B_i)$;
\item $\neg\neg(A,B) = (A,B)$, the famous [[double negation]] law.

\end{itemize}
We can go on to define the relative complement $(A,B) \setminus (C,D)$ and symmetric difference $(A,B) \uplus (C,D)$ in terms of complements, intersections, and unions as usual, and they obey many of the usual classical laws. (For instance, $\uplus$ is ---through a fairly lengthy calculation--- associative, which is not constructively true of [[symmetric difference]] on a power set.)

At this point, the reader could be forgiven for thinking that we have cleverly pulled a [[Boolean algebra]] out of a mere [[Heyting algebra]], but this is not true; aside from the give-away that this lattice is not constructively completely distributive, it is not even classically a Boolean algebra. This is because $(A,B) \cup \neg(A,B) = (A \cup B, \empty)$ (and similarly for intersection) and there is no requirement that $A \cup B = X$. What we have instead is a complete Boolean [[rig]], aka semi-ring with unit; to keep consistent with the usual terminology of measure theory, I'll call such a thing a \textbf{Boolean semi-algebra}.

This is all a special case of the [[Chu construction]]; the poset of disjoint pairs in $X$ is $Chu_{TV}(P X, \empty)$, where $TV$ is the [[enriched category|enriching]] category of [[truth values]].

\hypertarget{the_semialgebra_of_complemented_pairs}{}\subsubsection*{{The $\sigma$-semi-algebra of complemented pairs}}\label{the_semialgebra_of_complemented_pairs}

Given a set $X$, a \textbf{$\sigma$-semi-algebra} on $X$ is a collection $\mathcal{M}$ of disjoint pairs in $X$, called \textbf{complemented pairs}, such that: 1. $[\empty] = (\empty,X)$ is a complemented pair; 2. If $(A,B)$ is a complemented pair, then so is its complement $(B,A)$; 3. If $(A_1,B_1), (A_2,B_2), (A_3,B_3), \ldots$ are complemented pairs, then so is their union $(\bigcup_i A_i, \bigcap_i B_i)$.

The arguments above that $\mathcal{M}$ is closed under countable intersections, relative complements, and symmetric differences goes through. (We can also define analogous notions of semi-algebra, $\delta$-semi-ring, and other variations of $\sigma$-[[sigma-algebra|algebra]].)

Finally, a \textbf{Cheng measurable space} or simply a \textbf{Cheng space} is a set equipped with a $\sigma$-semi-algebra.

(Incidentally, the reason why we do not use the term `measurable pair' is that $A$ and $B$ may easily both be measurable in some sense yet without having $(A,B)$ as a complemented pair. In particular, $(\empty,\empty)$ is rarely a complemented pair ---although that is not forbidden either---, yet it is hard to call it non-measurable.)

\hypertarget{measurable_sets_and_functions}{}\subsubsection*{{Measurable sets and functions}}\label{measurable_sets_and_functions}

A [[subset]] $A$ of a Cheng space $X$ is \textbf{[[measurable subset|measurable]]} if there is some complemented pair $(A,B)$. A subset $S$ is \textbf{[[full subset|full]]} if, for some complemented pair $(A,B)$, $S$ contains the [[union]] $A \cup B$. Conversely, $S$ is \textbf{[[null subset|null]]} if, for some such $(A,B)$, $S$ is disjoint from $A \cup B$.

Given two Cheng measurable spaces $X$ and $Y$, an \textbf{[[almost function]]} from $X$ to $Y$ is a [[partial function]] from $X$ to $Y$ such that the [[domain]] of $f$ is full. An almost function is \textbf{[[measurable function|measurable]]} if, given any complemented pair $(C,D)$ in $Y$, there is a complemented pair $(A,B)$ such that $\{p\colon X \;|\; p \in A \iff f(p) \in C\}$ and $\{p\colon X \;|\; p \in B \iff f(p) \in D)$ are both full.

Two (measurable) functions are \textbf{[[almost equal]]} if their [[equaliser]] is full.

Cheng spaces, measurable almost functions between them, and almost equality between them form a [[category]] $Cheng Sp$.

\hypertarget{completion}{}\subsection*{{Completion}}\label{completion}

A Cheng space is \textbf{[[complete measure space|complete]]} if, whenever $(A,B)$ is a complemented pair and $A \Leftrightarrow C$ and $B \Leftrightarrow D$ are full, then $(C,D)$ is a complemented pair. In particular, every full set and every null set is measurable.

Given any Cheng space $(X,\mathcal{M})$, its \textbf{completion} $(X,\bar{\mathcal{M}})$ has the same [[underlying set]] $X$ but a disjoint pair $(C,D)$ is $\bar{\mathcal{M}}$-complemented iff, for some $\mathcal{M}$-complemented pair $(A,B)$, both $A \Leftrightarrow C$ and $B \Leftrightarrow D$ are $\mathcal{M}$-full.

A Cheng space is complete iff it is its own completion; the completion of any Cheng space is complete. The [[identity function]] on $X$ is measurable in either direction between $(X,\mathcal{M})$ and $(X,\bar{\mathcal{M}})$, so they are [[isomorphic]] in $Cheng Sp$. Accordingly, the [[full subcategory]] of $Cheng Sp$ on the complete spaces is [[equivalence of categories|equivalent]] to $Cheng Sp$ itself (via its [[inclusion functor]]).

When restricted to complete spaces, the definition of measurable function is simpler: any partial function under which the [[preimage]] of any complemented pair is complemented.

\hypertarget{measures_on_cheng_spaces}{}\subsection*{{Measures on Cheng spaces}}\label{measures_on_cheng_spaces}

We will define the notion of a [[positive measure]] on a Cheng space; it's straightforward to generalise to other sorts of measures as described at [[measure]].

Given a Cheng space $(X,\mathcal{M})$, a \textbf{positive measure} on $(X,\mathcal{M})$ is a [[function]] $\mu$ from $\mathcal{M}$ to $[0,\infty]$ such that:

\begin{itemize}%
\item [[absolutely continuous measure|absolute continuity]]: $\mu(A,B) = 0$ if $B$ is full;
\item [[inclusion-exclusion]]: $\mu(A \cap C, B \cup D) + \mu(A \cup C, B \cap D) = \mu(A,B) + \mu(C,D)$;
\item [[subadditive function|subadditivity]]: $\mu(\bigcup_i A_i, \bigcap_i B_i) \leq \sum_i \mu(A_i,B_i)$ for an [[infinite sequence]] of complemented pairs.

\end{itemize}
We think of $\mu(A,B)$ as the measure of $A$; thanks to absolute continuity, either $A$ or $B$ alone is enough to determine $\mu(A,B)$.

[[!redirects Cheng space]] [[!redirects Cheng spaces]] [[!redirects Cheng measurable space]] [[!redirects Cheng measurable spaces]]

[[!redirects boolean semialgebra]] [[!redirects boolean semialgebras]] [[!redirects boolean semi-algebra]] [[!redirects boolean semi-algebras]] [[!redirects Boolean semialgebra]] [[!redirects Boolean semialgebras]] [[!redirects Boolean semi-algebra]] [[!redirects Boolean semi-algebras]]



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