nLab sober measurable space

Idea

Sober measurable spaces are to measurable spaces as sober topological spaces are to topological spaces.

More in detail, a measurable space is sober if and only if every zero-one measure on it is a Dirac delta at a unique point. Equivalently, if its sigma-algebra separates points, and if every irreducible measure is induced by a point. (See below.)

This allows to draw parallels between measure theory and point-free topology.

Definition

A measurable space $X$ is called sober if any of the following equivalent conditions hold:

• Every zero-one measure on $X$ is the Dirac delta over a unique point.
• Every zero-one kernel into $X$ is induced by a unique measurable function.
• The component $\eta_S:X\to S X$ of the unit of the zero-one measure monad is an isomorphism.
• The following diagram of Meas is an equalizer,

where $P$ denotes the Giry monad and $\delta$ is its unit, given by Dirac deltas. (Note that the diagram is always a fork, by naturality of $\delta$.)

The last condition is sometimes called the equalizing requirement (Moggi’91).

Analogy with the topological case

Compare the definition with the following equivalent conditions in order for a topological space to be sober:

• Every completely prime filter (or classically equivalently, irreducible closed set, point of a locale) is induced by a unique point.
• Every morphism of locales into $X$ is induced by a unique continuous function.
• The component at $X$ of the unit of the sobrification monad is an isomorphism.
• The following diagram of Top is an equalizer,

where $V$ denotes the monad of closed subsets (with the lower Vietoris topology?) and $\sigma$ is its unit, given by (closures of) singletons.

Examples

• Every standard Borel space is sober.
• Given any measurable space $X$, the space of probability measures $P X$ (Giry monad) is always sober.
• The invariant sigma algebra of an ergodic dynamical system is in general not sober.

Categorical properties

Sober measurable spaces form a reflective subcategory of Meas. That is, the inclusion functor $SoberMeas\hookrightarrow Meas$ has a left adjoint $S:Meas\to SoberMeas$. It maps a measurable space $X$ to the space of zero-one measures over it, which is equivalently given by the following equalizer, where $P$ denotes the Giry monad and $\delta$ is its unit.

Analogously to the case of topological spaces, this functor is sometimes called the sobrification.

To see its universal property, notice that by naturality of $\delta$, and by definition of equalizer, there is a unique map $X\to S X$ making the triangle below commute. This map $X\to SX$ forms the unit of the adjunction.

(More on this at zero-one measure.)

Separating points and having enough points

A sigma-algebra $\mathcal{A}$ on a set $X$ is said to separate points if any of the following equivalent conditions holds:

• Given $x,y\in X$, $x=y$ if and only if for all $A\in\mathcal{A}$, $x\in A$ if and only if $y\in A$.
• Given $x,y\in X$, $x=y$ if and only if for every measurable function $f:(X,\mathcal{A})\to\mathbb{R}$, $f(x)=f(y)$.
• Given $x,y\in X$, $x=y$ if and only if for every Markov kernel $k:(X,\mathcal{A})\to (Y,\mathcal{B})$ and every $B\in\mathcal{B}$, $k(B|x)=k(B|y)$.
• The Dirac delta map (unit of the Giry monad) $\delta:X\to P X$, $x\mapsto\delta_x$ is injective.

This condition is analogous to the T0 (Kolmogorov) separation axiom for topological spaces: it says that the sigma-algebra can distinguish any two points.

One can also define an equivalence relation analogous to the specialization preorder, as follows:

$$x\sim y \qquad\Leftrightarrow\qquad \forall A\in\mathcal{A}, x\in A \Rightarrow y\in A .$$

(Note that the relation is symmetric because sigma-algebras, unlike topologies, are closed under complements.) A sigma-algebra separates points precisely if this relation is discrete.

Every sober measurable space has a sigma-algebra which separates points. Just as for topological spaces, however, separating points is not enough: one also needs a condition of having enough points.

Definition

We say that a measurable space $(X,\mathcal{A})$ has enough points if and only if every zero-one measure is a Dirac measure.

This way, a measurable space is sober if and only if it separates points and it has enough points.

Here is another analogy with topological spaces: the sobrification process can be taken in two steps: First, we quotient under the relation of “being indistinguishable for the sigma-algebra” (analogous to the Kolmogorov quotient). Then we “add the missing points”.

(Compare again with the topological case.)

Related entries

• zero-one kernel, Markov kernel, Stoch
• sober topological space, locale, point of a locale, frame, point-free topology
• Markov category, probability monad

References

• Sean Moss, Paolo Perrone, Probability monads with submonads of deterministic states, LICS 2022. (arXiv:2204.07003)

• Anna Bucalo and Giuseppe Rosolini, Sobriety for equilogical spaces. Theorerical Computer Science 546, 2014. (doi)

• Paul Taylor, Sober spaces and continuations. Theory and Applications of Categories 10(12), 2002. (link)

• Eugenio Moggi, Notions of computations and monads, Information and Computation 93(1), 1991 (LICS 1989).

Introductory material

• Paolo Perrone, When measurable spaces don’t have enough points, introductory talk. (YouTube)

category: probability