Measurable locales are certain locales, which may serve as a basis for measure theory much as general locales serve as a basis for topology (especially in constructive mathematics). Specifically, a measurable locale is equivalent to a localisable measurable space (or dual to a commutative von Neumann algebra).
Ironically, in constructive mathematics, measurable locales are not locales (except for the empty space), on pain of excluded middle.
The concept appears to be due to Dmitri Pavlov.
At present, there is no purely order-theoretic definition of measurable locales. However, there are a few other ways of defining them.
First we give definitions appropriate for classical mathematics.
As a preliminary step, consider the category $Comp Bool Alg_{sup}$ of complete boolean algebras and supremum-preserving homomorphisms of boolean algebras. This is a full subcategory of the category $Frm$ of frames and so its opposite category $Comp Bool Alg_{sup}^{op}$ is a full subcategory of the category Loc of locales. The category $Meas Loc$ of measurable locales is yet further a full subcategory of $Comp Bool Alg_{sup}^{op}$, so our job is simply to specify which complete boolean algebras are the objects of this category. There are at least three equivalent ways of doing so.
By one definition, a complete boolean algebra $L$ is a measurable locale if there is a complete measure space $X$ such that $L$ is isomorphic (as a boolean algebra) to the boolean algebra $\mathcal{M}/\mathcal{N}$ of measurable subsets of $X$ modulo null subsets. Note that the measure on $X$ is irrelevant except to specify the null subsets. In this way, $Meas Loc$ becomes equivalent to the category $Loc Meas$ of localisable measurable spaces. (Not every measure space has a complete boolean algebra as $\mathcal{M}/\mathcal{N}$; those which do are called localisable, and we interpret the term ‘localisable measurable space’ to refer to a structure with both $\mathcal{M}$ and $\mathcal{N}$ specified.)
By another definition, a complete boolean algebra $L$ is a measurable locale if for every element $x \neq 0$ of $L$ there is a normal measure? (see below) $\mu$ on $L$ valued in $[0,\infty]$ such that $\mu(x) = 1$. (It would be enough to demand that $0 \lt \mu(x) \lt \infty$, since we can rescale the measure.)
By yet another definition, a complete boolean algebra $L$ is a measurable locale if there is a commutative $W^*$-algebra $A$ such that $L$ is isomorphic (as a boolean algebra) to the boolean algebra $Proj(A)$ of projection operators in $A$; (see idempotent operator for a construction of $Proj(A)$). In this way, $Meas Loc$ becomes dual to the category $Comm W^* Alg$ of commutative $W^*$-algebras. For this definition, we need not require that $L \cong Proj(A)$ be complete (or even a boolean algebra); this can be proved.
Whichever of these equivalent definitions is adopted, a measurable function between measurable locales is simply a continuous map between them as locales; these are the morphisms of $Meas Loc$. In more elementary terms, a measurable function from $L$ to $M$ (thought of as measurable locales) is a function from $M$ to $L$ (thought of as Boolean algebras) that preserves all joins (and then it also preserves finite meets and so is a frame homomorphism).
(This section is not due to Pavlov.)
For purposes of constructive mathematics, Toby Bartels suspects that it is appropriate to use the definition from $W^\star$-algebras, so long as we allow norms to be lower real numbers. (If they are all located, then $Proj(A)$ has decidable equality, which we don't want to require.) The other definitions seem more difficult to work with (primarily because they require completeness explicitly).
For this definition (as was remarked above), we need not require that $L \cong Proj(A)$ be complete. Constructively, we cannot require this; $Proj(A)$ need not be complete (although it is still a boolean algebra). Indeed, consider the point (see the examples), based on $A \coloneqq \mathbb{C}$, which is not the complete Heyting algebra of all truth values but only the (possibly incomplete) boolean algebra $\{\bot, \top\}$ of classical truth values (corresponding to the self-adjoint idempotent complex numbers $0$ and $1$).
It seems that there is some notion of completeness that applies here; $L$ should in some sense be ‘measurably complete’. In the point, for example, the subsingleton $\{* \;|\; P\}$, where $P$ is a truth value, is measurable iff $P$ is true or false, so the supremum of $\{\top \;|\; P\}$ exists in $L$ under the same circumstances. Figuring this out could allow the definitions through measurable spaces or normal measures to work.
The empty space, which is initial in $Meas Loc$, is the terminal boolean algebra with one element.
The point, which is terminal in $Meas Loc$, is the initial boolean algebra $\{\bot, \top\}$ of (classical) truth values.
The real line is the boolean algebra of Lebesgue-measurable sets of real numbers modulo the null sets. This is complete as a boolean algebra because … [argument needed] (and even constructively, it is still a boolean algebra).
Applying the classification of $W^*$-algebras, we find that (up to isomorphism), every measurable locale is a disjoint union (dually a direct product of boolean algebras) of points or of points and infinitely many real lines; a single real line is already isomorphic to the union of countably infinitely many real lines. (Of course, we can't expect this classification theorem? to hold constructively.)
To do measure theory, it's important to know how to interpret the real line with the Borel sets (rather than Lebesgue sets) as a measurable space. This is actually a little tricky, because Lebesgue measure is not complete on the Borel sets, and passing to the Lebesgue sets gives a different measurable space. We might simply take the Borel sets as they are (so that only the empty set is null), but then this is not complete as a boolean algebra.
One might suspect that there is no Borel real line in $Meas Loc$, which would cast serious doubts on that as a category for measure theory. But we can argue by abstract nonsense that it must exist, and more generally that any locale (including the locale of opens of any topological space) has a Borel measurable locale, as follows:
… http://mathoverflow.net/questions/31603/why-do-probabilists-take-random-variables-to-be-borel-and-not-lebesgue-measurab/31724#31724 … representable functor theorem …
In the case of the real line, the resulting measurable locale is actually rather complicated: something like an uncountable union of points and lines.
This process is a functor $B\colon Loc \to Meas Loc$, which we can also extend to $Top \to Loc \to Meas Loc$.
Similarly, smooth manifolds give rise to (much simpler!) measurable locales by taking the Lebesgue sets (possible since we know which sets are null sets on a smooth manifold). This process is a functor $L\colon SmoothMan_subm \to MeasLoc$, where the morphisms in $SmoothMan_subm$ are the smooth submersions.
In the case of the real line, we get the usual line described under the basic examples above. In fact, any second-countable smooth manifold with positive dimension gives rise to this measurable locale; the really interesting thing about $L$ is what it does to the morphisms.
On any complete boolean algebra $L$, given any abelian monoid $R$ equipped with a convergence structure (such as $[0,\infty]$), a normal measure? on $L$ valued in $R$ is a function $\mu\colon L \to R$ such that:
additivity: $\mu(\bot) = 0$ and $\mu(x \wedge y) + \mu(x \vee y) = \mu(x) + \mu(y)$;
continuity?: if $S$ is a downward-directed subset of $L$ whose infimum is the bottom element, then the net $(\mu(x))_{x\colon S}$ converges to $0$.
One could, of course, define a garden-variety measure by requiring continuity only when $S$ is the image of a decreasing sequence, but apparently normal measures are what we want for measurable locales.
(Perhaps there is a theorem that absolutely continuous measures on, say, Lebesgue space correspond to normal measures on the real line as a measurable locale? Certainly Lebesgue measure is not normal on the $\sigma$-algebra of measurable sets, but presumably it is normal on the boolean algebra of measurable sets modulo null sets, which is what we want.)
I'm not entirely sure how this works in constructive mathematics, but $[0,\infty]$ (for a positive measure) would be the space of nonnegative lower reals.
The theory of measurable locales seems to be published entirely on MathOverflow, in various questions and answers by Dmitri Pavlov. Here is an index:
Another post not in this index is: