nLab zero-one measure

Redirected from "zero-one measures".
Contents

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Measure and probability theory

Contents

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 pp on a measurable space XX 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) AA of XX,
    p(A)=0orp(A)=1. p(A) \;=\; 0 \qquad or \qquad p(A) \;=\; 1 .
  • Every real-valued random variable ff on the probability space (X,p)(X,p) almost surely constant (i.e. it is a determinstic random variable?).
  • Every event (measurable subset) AA of XX is independent of itself:
    p(A)=p(AA)=p(A)p(A). p(A) \;=\; p(A\cap A) \;=\; p(A)\,p(A) .
  • Every two events A,BA,B of XX are independent:
    p(AB)=p(A)p(B). p(A\cap B) \;=\; p(A)\,p(B) .
  • pp cannot be expressed as a nontrivial convex combination of other probability measures: if
    p=λq 1+(1λ)q 2 p \;=\; \lambda\,q_1 + (1-\lambda)\,q_2

    for λ(0,1)\lambda\in(0,1) and q 1,q 2PXq_1,q_2\in P X, then q 1=q 2=qq_1=q_2=q. (This can be seen as analogous to irreducible closed sets.)

  • For every f,g:Xf,g:X\to\mathbb{R} for which the integrals below exist, integration with pp preserves multiplication:
    X(fg)dp=( Xfdp)( Xgdp). \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 pp. (PP denotes the Giry monad, and δ\delta is its unit, given by Dirac measures.)

Examples

Further properties

The monad of zero-one measures

Zero-one measures form a monad, which we denote by (S,η S,μ S)(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 TopTop, 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 XX be a measurable space, and denote its sigma-algebra by Σ X\Sigma_X.

Functor and unit

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

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

    for all measurable AXA\subseteq X.

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

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

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

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

Proposition

The assignment AA *A\mapsto A^* is an isomorphism of sigma-algebras Σ XΣ SX\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 πSSX\pi\in S S X, we take μ S(π)SX\mu_S(\pi)\in S X to be defined by

μ S(π)(A)π(A *). \mu_S(\pi)(A) \;\coloneqq\; \pi(A^*) .

Notice that for every pSXp\in S X and πinSSX\pi in S S X,

μ S(δ S(p))(A)=δ S(p)(A *)=1 A *(p)=p(A) \mu_S(\delta_S(p))(A) \;=\; \delta_S(p)(A^*) \;=\; 1_{A^*}(p) \;=\; p(A)

and

δ S(μ S(π))(A *)=1 A*(μ S(π))=μ S(π)(A)=π(A *). \delta_S(\mu_S(\pi))(A^*) \;=\; 1_{A*}(\mu_S(\pi)) \;=\; \mu_S(\pi)(A) \;=\; \pi(A^*) .

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

Kleisli category

The Kleisli category of the monad SS is the category of measurable spaces and zero-one kernels, sometimes denoted by Stoch detStoch_det.

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

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

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

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

    for all BΣ YB\in\Sigma_Y, where

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

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

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

ϵ S(A|p)=p(A) \epsilon_S(A|p) \;=\; p(A)

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

Proposition

The kernel ϵ:SYY\epsilon: S Y \to Y is an isomorphism of Stoch detStoch_det, where its inverse is induced by the function η S:XSX\eta_S:X\to S X.

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

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

category: probability

Last revised on July 18, 2024 at 09:51:16. See the history of this page for a list of all contributions to it.