nLab zero-one measure

Idea

Zero-one measures are measures whose only values are zero and one.

In probability theory, they model situations which “are not really random”, where we are almost surely certain of which events take place and which do not. They are used to express zero-one laws and more generally to model situations of ergodicity.

Zero-one measures form a monad, which is analogous to sobrification of topological spaces.

The analogous concept for Markov kernels is a zero-one kernel.

Definition

A probability measure $p$ on a measurable space $X$ is said to be zero-one, irreducible or extremal if and only if any of the following equivalent conditions hold:

• For every event (measurable subset) $A$ of $X$,
  $$p(A) \;=\; 0 \qquad or \qquad p(A) \;=\; 1 .$$
• Every real-valued random variable $f$ on the probability space $(X,p)$ is almost surely constant (i.e. it is a determinstic random variable?).
• Every event (measurable subset) $A$ of $X$ is independent of itself:
  $$p(A) \;=\; p(A\cap A) \;=\; p(A)\,p(A) .$$
• Every two events $A,B$ of $X$ are independent:
  $$p(A\cap B) \;=\; p(A)\,p(B) .$$
• $p$ cannot be expressed as a nontrivial convex combination of other probability measures: if
  $$p \;=\; \lambda\,q_1 + (1-\lambda)\,q_2$$

  for $\lambda\in(0,1)$ and $q_1,q_2\in P X$, then $q_1=q_2=q$. (This can be seen as analogous to irreducible closed sets.)

• For every $f,g:X\to\mathbb{R}$ for which the integrals below exist, integration with $p$ preserves multiplication:
  $$\int_X (f\cdot g)\,dp \;=\; \left( \int_X f\,dp \right) \cdot \left( \int_X g\,dp \right) .$$
• The parallel pair of maps

agree on $p$. ($P$ denotes the Giry monad, and $\delta$ is its unit, given by Dirac measures.)

Examples

• Every Dirac delta measure is zero-one:

  $$\delta_x(A) \;=\; 1_A(x) \;=\; \begin{cases} 1 & x\in A ; \\ 0 & x\notin A . \end{cases}$$

  If $X$ is standard Borel, or more more generally if it has enough points, every zero-one measure on $X$ is a Dirac delta.

• Every ergodic measure is zero-one when restricted to the invariant sigma-algebra.

Further properties

• Zero-one measures are exactly the laws of those random variables which satisfy a zero-one law.

• Zero-one kernels are exactly the deterministic states of Stoch, in the sense of Markov categories.

The monad of zero-one measures

Zero-one measures form a monad, which we denote by $(S,\eta_S,\mu_S)$, a submonad of the Giry monad. It can be seen as an analogue of the monad of completely prime filters (sobrification) on $Top$, and because of that we also call it the sobrification monad of measurable spaces (see also at sober measurable space). This monad is defined abstractly in MP’22 (based on previous work, see the references), here we give a more explicit description.

In what follows, let $X$ be a measurable space, and denote its sigma-algebra by $\Sigma_X$.

Functor and unit

The space of zero-one measures $S X$ on $X$ can be equivalently described as follows:

• It is the equalizer of the following pair:

• It is the subset of $P X$ (the Giry monad) of those measures which are zero-one, with the sigma-algebra induced from the one of $P X$;
• It is the set of all zero-one measures on $X$, together with the sigma-algebra $\Sigma_{S X}$ given by the following sets,
  $$A^* \;\coloneqq\; \{p\in S X \;:\; p(A) = 1 \}$$

  for all measurable $A\subseteq X$.

The unit of the monad can be characterized equivalently as follows:

• Since every Dirac measure is zero-one, the map $\delta:X\to P X$ lands in $S X$.
• Notice that by naturality of $\delta$ (the unit of the Giry monad), the map $\delta:X\to P X$ factors uniquely through the equalizer $S X$:

We take as unit $\eta_S$ the resulting map $X\to S X$. Explicitly,

$$\eta_S(x)(A) \;=\; 1_A(x) \;=\; \begin{cases} 1 & x\in A ; \\ 0 & x\notin A . \end{cases}$$

Proposition

The assignment $A\mapsto A^*$ is an isomorphism of sigma-algebras $\Sigma_X\to \Sigma_{S X}$. Its inverse is given by the preimage map

(Compare to how a topological space and its sobrification have the same frame.)

Multiplication and idempotency

We can define the multiplication of the monad, explicitly, as follows. Given a zero-one measure over zero-one measures $\pi\in S S X$, we take $\mu_S(\pi)\in S X$ to be defined by

$$\mu_S(\pi)(A) \;\coloneqq\; \pi(A^*) .$$

Notice that for every $p\in S X$ and $\pi in S S X$,

$$\mu_S(\delta_S(p))(A) \;=\; \delta_S(p)(A^*) \;=\; 1_{A^*}(p) \;=\; p(A)$$

and

$$\delta_S(\mu_S(\pi))(A^*) \;=\; 1_{A*}(\mu_S(\pi)) \;=\; \mu_S(\pi)(A) \;=\; \pi(A^*) .$$

Therefore (using the fact that $A\mapsto A^*$ is a bijection), $\delta_S: S X\to S S X$ and $\mu_S: S S X\to S X$ are mutually inverse. This makes $S$ an idempotent submonad of the Giry monad.

Kleisli category

The Kleisli category of the monad $S$ is the category of measurable spaces and zero-one kernels, sometimes denoted by $Stoch_det$.

Since the monad is idempotent, its Kleisli category is equivalent to its Eilenberg-Moore category, which is the category $SoberMeas$ of sober measurable spaces.

Let’s now write this equivalence explicitly. One one side, define the functor $F:SoberMeas\to Stoch_det$ as follows:

• On objects, it is the identity (or better said, it is forgetful from sober measurable spaces to all measurable spaces).
• On morphisms, it maps a measurable function $f:X\to Y$ to the zero-one kernel $\delta_f:X\to Y$ induced by $f$.

Similarly we can define a functor $G:Stoch_det\to SoberMeas$ as follows:

• On objects, it maps a (not necessarily sober) measurable space $X$ to $S X$.
• On morphisms, it maps a zero-one kernel $k:X\to Y$ to the map $S X \to S Y$ which takes a zero-one measure $p\in S X$ and gives the zero-one measure on $Y$ defined by
  $$B \mapsto p(k^*B) ,$$

  for all $B\in\Sigma_Y$, where

  $$k^*B \;\coloneqq\; \{x\in X \,:\, k(B|x) = 1 \} .$$

The unit $\eta_S:X\to S X$ induces a natural transformation with components $X\to GFX$. When $X$ is sober, this component is an isomorphism.

Conversely, for a generic measurable space $Y$ (not necessarily sober), we have a natural zero-one kernel $\epsilon_S: S Y \to Y$ defined by

$$\epsilon_S(A|p) \;=\; p(A)$$

for all $p\in S Y$ and all measurable subsets $A\subseteq Y$. (This can be seen as the restriction of the sampling map to zero-one measures.)

Proposition

The kernel $\epsilon: S Y \to Y$ is an isomorphism of $Stoch_det$, where its inverse is induced by the function $\eta_S:X\to S X$.

This way we have natural isomorphisms $\eta_S:id\Rightarrow G\circ F$ and $\epsilon_S:F\circ G\Rightarrow id$, which give an equivalence of categories.

Related entries

• zero-one kernel
• Markov kernel, Markov category, Stoch, Krn
• zero-one law, ergodicity, deterministic random variable
• thunk-force category
• sober measurable space
• point-free topology, point of a locale

References

• Tobias Fritz, A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. Adv. Math., 370:107239, 2020. arXiv:1908.07021.

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

• Sean Moss, Paolo Perrone, A category-theoretic proof of the ergodic decomposition theorem, Ergodic Theory and Dynamical Systems, 2023. (arXiv:2207.07353)

• Noé Ensarguet, Paolo Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, arXiv:2310.04267

• 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