nLab Loomis-Sikorski duality

Idea

Loomis-Sikorski duality is a Stone-type duality between measurable spaces and particular (σ-complete) Boolean algebras, and it is one of the main approaches to point-free measure theory?.

It is closely related to point-free topology, as well as to measurable Gelfand duality? and to the duality between measure spaces (not measurable) and measurable locales.

Main concepts

Recall that a measurable space is a set $X$ equipped with a σ-algebra $\Sigma$ of subsets of $X$. These subsets have the interpretation of “distinctions” that one can make, analogously to the open sets of a topology. Moreover, in the presence of a measure, these are precisely the subsets which have a “size” (the value of the measure), and in probability theory, they are the events to which one assigns a probability.

In point-free measure theory?, just as in point-free topology, one is interested in shifting the focus from the points of $X$ to the ordered structure of $\Sigma$, and to study to which extent they determine one another.

From measurable spaces to σ-complete Boolean algebras

Given a measurable space $(X,\Sigma_X)$, the σ-algebra $\Sigma_X$ is by definition closed under complements and countable unions. In other words it is a σ-complete Boolean algebra (also known as Boolean σ-algebra).

Given a measurable map $f:(X,\Sigma_X)\to (Y,\Sigma_Y)$, the inverse image map preserves countable unions and complements, we therefore have a functor from measurable spaces to σ-complete Boolean algebras (where the morphisms are required to preserve countable unions and complements).

From σ-complete Boolean algebras to measurable spaces

Let $\Sigma$ be a σ-complete Boolean algebra. We can define a point of $\Sigma$ to be any of the following equivalent structures:

• A zero-one measure on $\Sigma$, i.e. an assignment $p:\Sigma\to\{0,1\}$ which is countably additive and for which $p(X)=1$;
• An assignment $p:\Sigma\to\{0,1\}$ which is a morphism of σ-complete Boolean algebras;
• A prime σ-filter on $\Sigma$, i.e. a proper subset of $\Sigma$ which is closed under countable joins and meets;
• A σ-ultrafilter on $\Sigma$, i.e. an ultrafilter on $\Sigma$ which is closed under countable meets.

Given a morphism $\phi:\Sigma\to\Gamma$ and a point $p$ of $\Gamma$, the composition $p\circ\phi$ is a point of $\Sigma$. Denoting by $Sp(\Sigma)$ the space of points of $\Sigma$ (or spectrum), we then have a function $\phi^*:Sp(\Gamma)\to Sp(\Sigma)$.

In order to turn this assignment into a functor $\sigma Bool^\mathrm{op}\to Meas$, we need to equip the spaces $Sp(\Sigma)$ with σ-algebras. We do it as follows (see also at zero-one measure - the monad): for every $A\in \Sigma$, define

$$A^* = \{p\in Sp(\Sigma) : p(A)=1 \} .$$

The sets $A^*$ form a σ-algebra, and the maps $\phi^*:Sp(\Gamma)\to Sp(\Sigma)$ are measurable. This gives the desired functor

The adjunction

Just as in the topological case, the functors $\Sigma: Meas^{op} \to \sigma Bool$ and $Sp: \sigma Bool^{op}\to Meas$ form a (contravariant) idempotent adjunction:

• The unit has as components measurable functions $\eta: X \to Sp(\Sigma_X)$ which map a point $x\in X$ to the corresponding principal filter? on $\Sigma_X$ (or equivalently, to the Dirac delta zero-one measure) $\delta_x$: for every $A\in\Sigma_X$,
  $$\delta_x (A) = \begin{cases} 1 & x\in A \\ 0 & x\notin A . \end{cases}$$
• The counit has as components morphisms of $\sigma Bool^{op}$ in the form $\epsilon^{op}\Sigma_{Sp(\Sigma)}\to\Sigma$, or equivalently, morphisms of $\sigma Bool$ of the form $\epsilon:\Sigma\to \Sigma_{Sp(\Sigma)}$ mapping $A\in\Sigma$ to the set $A^*\in\Sigma_{Sp(\Sigma)}$ defined above.

The adjunction induces an idempotent monad on $Meas$ (the monad of zero-one measures), and an idempotent monad on $\sigma Bool$ as well (or equivalently, an idempotent comonad on $\sigma Bool^{op}$).

As usual with idempotent adjunctions, we have that the center is equivalently given by

• The full subcategory of $Meas$ whose objects are those spaces for which the unit $\eta:X\to Sp(\Sigma_X)$ is an isomorphism. These are sometimes called sober measurable spaces. Analogously to sober topological spaces, these are the spaces whose points are uniquely determined by the σ-algebra.
• The full subcategory of $\sigma Bool$ whose objects are those σ-algebras for which the counit $\epsilon:\Sigma\to \Sigma_{Sp(\Sigma)}$ is an isomorphism. These are sometimes called spatial or concrete σ-algebras.

Because of this, sometimes one calls the monad on $Meas$ the sobrification monad.

Particular results

(Work in progress. For now see the references.)

See also

• sober measurable space, zero-one measure, zero-one kernel
• von Neumann algebra, measure algebra?
• σ-complete Boolean algebra, monotone-complete $C^*$-algebra?
• point-free measure theory?, measurable Gelfand duality?
• Stone duality, Gelfand duality, Isbell duality
• categorical approaches to probability theory

References

• Roman Sikorski, Boolean Algebras, Springer, 1969.

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

• Ruiyuan Chen, A universal characterization of standard Borel spaces. The Journal of Symbolic Logic, 88(2), 2023. (arXiv)

• Tobias Fritz and Antonio Lorenzin, Categories of abstract and noncommutative measurable spaces, 2025. (arXiv)

Introductory material

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

category: probability