Nonatomic Probability Space Research Articles

Embedding universality for II1 factors with property (T)

We prove that every separable tracial von Neumann algebra embeds into a II1 factor with property (T) which can be taken to have trivial outer automorphism and fundamental groups. We also establish an analogous result for the trivial extension over a non-atomic probability space of every countable p.m.p. equivalence relation. In addition, we obtain two new results concerning the structure of infinitely generic II1 factors. These results are obtained by using the class of wreath-like product groups introduced recently in [8].

Best Constants in Weighted Estimates for Dyadic Shifts

We identify the weighted $$L^p$$ -norms of shift operators in the context of nonatomic probability spaces equipped with tree-like structures.

On differentiability in the Wasserstein space and well-posedness for Hamilton–Jacobi equations

In this paper we elucidate the connection between various notions of differentiability in the Wasserstein space: some have been introduced intrinsically (in the Wasserstein space, by using typical objects from the theory of Optimal Transport) and used by various authors to study gradient flows, Hamiltonian flows, and Hamilton–Jacobi equations in this context. Another notion is extrinsic and arises from the identification of the Wasserstein space with the Hilbert space of square-integrable random variables on a non-atomic probability space. As a consequence, the classical theory of well-posedness for viscosity solutions for Hamilton–Jacobi equations in infinite-dimensional Hilbert spaces is brought to bear on well-posedness for Hamilton–Jacobi equations in the Wasserstein space.

Preferences over all random variables: Incompatibility of convexity and continuity

We consider preferences over all random variables on a given nonatomic probability space. We show that non-trivial and complete preferences cannot simultaneously satisfy the two fundamental principles of convexity and continuity. As an implication of this incompatibility result there cannot exist any non-trivial continuous utility representations over all random variables that are either quasi-concave or quasi-convex. This rules out standard risk-averse (or seeking) utility representations for this large space of random variables.

Compositions of conditional expectations, Amemiya–Andô conjecture and paradoxes of thermodynamics

We investigate properties of compositions of conditional expectations on a non-atomic probability space (Ω,F,μ). Let 1≤p<∞ and X,Y∈Lp(Ω,F,μ). If for any convex f:R→R we have Ef(X)≥Ef(Y), then for each ε>0 there exist conditional expectations P1,P2,P3 and E1,…,En∈{P1,P2,P3} such that ‖Y−En…E1X‖p<ε. This theorem seems particularly surprising if one simplifies the assumption by taking X and Y to have the same distribution since, intuitively, it seems to contradict the second law of thermodynamics. We use this theorem to build the following counterexample for the Amemiya–Andô conjecture: There exist f∈L1(R)∩L2(R), conditional expectations P1,…,P5 on (R,Borel(R),λ) and E1,E2,…∈{P1,…,P5} such that the sequence of iterations (En…E2E1f) diverges in L2(R).

Generic stationary measures and actions

Let G G be a countably infinite group, and let μ \mu be a generating probability measure on G G . We study the space of μ \mu -stationary Borel probability measures on a topological G G space, and in particular on Z G Z^G , where Z Z is any perfect Polish space. We also study the space of μ \mu -stationary, measurable G G -actions on a standard, nonatomic probability space. Equip the space of stationary measures with the weak* topology. When μ \mu has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of ( G , μ ) (G,\mu ) . When Z Z is compact, this implies that the simplex of μ \mu -stationary measures on Z G Z^G is a Poulsen simplex. We show that this is also the case for the simplex of stationary measures on { 0 , 1 } G \{0,1\}^G . We furthermore show that if the action of G G on its Poisson boundary is essentially free, then a generic measure is isomorphic to the Poisson boundary. Next, we consider the space of stationary actions, equipped with a standard topology known as the weak topology. Here we show that when G G has property (T), the ergodic actions are meager. We also construct a group G G without property (T) such that the ergodic actions are not dense, for some μ \mu . Finally, for a weaker topology on the set of actions, which we call the very weak topology, we show that a dynamical property (e.g., ergodicity) is topologically generic if and only if it is generic in the space of measures. There we also show a Glasner-King type 0-1 law stating that every dynamical property is either meager or residual.

A Relation between two Kinds of Norms for Martingales

Abstract Let X be a Banach function space over a nonatomic probability space Ω and let M- denote the collection of all uniformly integrable martingales on Ω. For f = (fn)n∈Z+ ∈Mu, let Mf denote the maximal function of f, and let f∞ denote the almost sure limit of f. We give some necessary and sufficient conditions for X to have the property that if f, g ∈Mu and ‖Mg‖x ≤ ‖Mf‖x, then ‖g∞‖X ≤ C‖f∞‖x.

On Doob’s inequality and Burkholder’s inequality in weak spaces

Let X be a Banach function space over a nonatomic probability space. We associate with X the weak space \({\mathrm {w}}\text {-}{X}\), which is a quasi-Banach space defined in a natural way. We give necessary and sufficient conditions for Doob’s inequality in \({\mathrm {w}}\text {-}{X}\) to be valid, and necessary and sufficient conditions for Burkholder’s inequality in \({\mathrm {w}}\text {-}{X}\) to be valid.

SHARP WEAK-TYPE ESTIMATES FOR THE DYADIC-LIKE MAXIMAL OPERATORS

We provide the explicit formula for the Bellman function corresponding to the weak-type $L^{p,\infty}\to L^{q,\infty}$ estimates for the dyadic maximal operator on $\mathbb{R}^n$. Actually, we do this in the more general setting of maximal operators associated with a tree-like structure on a nonatomic probability space. The approach rests on a clever combination of some novel optimization and combinatorial arguments.

(Uniform) convergence of twisted ergodic averages

Let$T$be an ergodic measure-preserving transformation on a non-atomic probability space$(X,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$. We prove uniform extensions of the Wiener–Wintner theorem in two settings: for averages involving weights coming from Hardy field functions $p$,$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(p(n))T^{n}f(x)\bigg\}; &amp; &amp; \displaystyle \nonumber\end{eqnarray}$$and for ‘twisted’ polynomial ergodic averages,$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(n\unicode[STIX]{x1D703})T^{P(n)}f(x)\bigg\} &amp; &amp; \displaystyle \nonumber\end{eqnarray}$$for certain classes of badly approximable$\unicode[STIX]{x1D703}\in [0,1]$. We also give an elementary proof that the above twisted polynomial averages converge pointwise$\unicode[STIX]{x1D707}$-almost everywhere for$f\in L^{p}(X),p&gt;1,$and arbitrary$\unicode[STIX]{x1D703}\in [0,1]$.

Minimal representation of insurance prices

This paper prices insurance contracts by employing law invariant, coherent risk measures from mathematical finance. We demonstrate that the corresponding premium principle enjoys a minimal representation. Uniqueness–in a sense specified in the paper–of this premium principle is derived from this initial result. The representations are derived from a result by Kusuoka, which is usually given for nonatomic probability spaces. We extend this setting to premium principles for spaces with atoms, as this is of particular importance for insurance.Further, stochastic order relations are employed to identify the minimal representation. It is shown that the premium principles in the minimal representation are extremal with respect to the order relations. The tools are finally employed to explicitly provide the minimal representation for premium principles, which are important in actuarial practice.

On some martingale inequalities for mean oscillations in weak spaces

Given a Banach function space \(X\) over a nonatomic probability space, we define, in a natural way, a quasi-Banach function space which is denoted by \(\mathrm {w}\text {-}{X}\). In this paper, we consider some inequalities in \(\mathrm {w}\text {-}{X}\) involving the mean oscillation and the maximal function of a martingale. The main results give necessary and sufficient conditions for the inequalities to be valid.

Kadison–Kastler stable factors

A conjecture of Kadison and Kastler from 1972 asks whether sufficiently close operator algebras in a natural uniform sense must be small unitary perturbations of one another. For n≥3 and a free, ergodic, probability measure-preserving action of SLn(Z) on a standard nonatomic probability space (X,μ), write M=(L∞(X,μ)⋊SLn(Z))⊗¯R, where R is the hyperfinite II1-factor. We show that whenever M is represented as a von Neumann algebra on some Hilbert space H and N⊆B(H) is sufficiently close to M, then there is a unitary u on H close to the identity operator with uMu∗=N. This provides the first nonamenable class of von Neumann algebras satisfying Kadison and Kastler’s conjecture. We also obtain stability results for crossed products L∞(X,μ)⋊Γ whenever the comparison map from the bounded to usual group cohomology vanishes in degree 2 for the module L2(X,μ). In this case, any von Neumann algebra sufficiently close to such a crossed product is necessarily isomorphic to it. In particular, this result applies when Γ is a free group.

Uniform boundedness of conditional expectation operators on a Banach function space

Let X be a Banach function space over a nonatomic probability space. The quasiBanach space weak–X is defined in a natural way. We give some necessary and sufficient conditions on X for all the conditional expectation operators to be uniformly bounded operators from X into weak–X . Mathematics subject classification (2010): 46E30, 47B38, 60G42.

Partition sensitivity for measurable maps

We study countable partitions for measurable maps on measure spaces such that, for every point $x$, the set of points with the same itinerary as that of $x$ is negligible. We prove in nonatomic probability spaces that every strong generator (Parry, W., Aperiodic transformations and generators, J. London Math. Soc. 43 (1968), 191–194) satisfies this property (but not conversely). In addition, measurable maps carrying partitions with this property are aperiodic and their corresponding spaces are nonatomic. From this we obtain a characterization of nonsingular countable-to-one mappings with these partitions on nonatomic Lebesgue probability spaces as those having strong generators. Furthermore, maps carrying these partitions include ergodic measure-preserving ones with positive entropy on probability spaces (thus extending the result in Cadre, B., Jacob, P., On pairwise sensitivity, J. Math. Anal. Appl. 309 (2005), 375–382). Some applications are given.

On characterization of a.s. convergence of all conditionings for sequences of random variables

Abstract We prove that in a non-atomic probability space, for a sequence of any integrable r.v. (Xn ) we have the following equivalence: 𝔼(Xn |𝒰) → 0 a.s. for any σ-field 𝒰 of events iff Xn → 0 a.s. and 𝔼 sup n ≥1 |Xn | &lt; ∞.

Weak mixing suspension flows over shifts of finite type are universal

Let $S$ be an ergodic measure-preserving automorphism on a nonatomicprobability space, and let $T$ be the time-one map of a topologically weakmixing suspension flow over an irreducible subshift of finite type under aHölder ceiling function. We show that if the measure-theoretic entropyof $S$ is strictly less than the topological entropy of $T$, then thereexists an embedding of the measure-preserving automorphism into thesuspension flow. As a corollary of this result and the symbolic dynamicsfor geodesic flows on compact surfaces of negative curvature developed byBowen [5] and Ratner [31], we also obtain anembedding of the measure-preserving automorphism into a geodesic flowwhenever the measure-theoretic entropy of $S$ is strictly less than thetopological entropy of the time-one map of the geodesic flow.

A note on the absurd law of large numbers in economics

Let Γ be a Borel probability measure on R and ( T , C , Q ) a nonatomic probability space. Define H = { H ∈ C : Q ( H ) > 0 } . In some economic models, the following condition is requested. There is a probability space ( Ω , A , P ) and a real process X = { X t : t ∈ T } satisfying for each H ∈ H , there is A H ∈ A with P ( A H ) = 1 such that t ↦ X ( t , ω ) is measurable and Q ( { t : X ( t , ω ) ∈ ⋅ } | H ) = Γ ( ⋅ ) for ω ∈ A H . Such a condition fails if P is countably additive, C countably generated and Γ nontrivial. Instead, as shown in this note, it holds for any C and Γ under a finitely additive probability P. Also, X can be taken to have any given distribution.

Iterated conditional expectations

Consider the probability space ( [ 0 , 1 ) , B , λ ) , where B is the Borel σ-algebra on [ 0 , 1 ) and λ the Lebesgue measure. Let f = 1 [ 0 , 1 / 2 ) and g = 1 [ 1 / 2 , 1 ) . Then for any ε > 0 there exists a finite sequence of sub- σ-algebras G j ⊂ B ( j = 1 , … , N ) , such that putting f 0 = f and f j = E ( f j − 1 | G j ) , j = 1 , … , N , we have ‖ f N − g ‖ ∞ < ε ; here E ( ⋅ | G j ) denotes the operator of conditional expectation given σ-algebra G j . This is a particular case of a surprising result by Cherny and Grigoriev (2007) [1] in which f and g are arbitrary equidistributed bounded random variables on a nonatomic probability space. The proof given in Cherny and Grigoriev (2007) [1] is very complicated. The purpose of this note is to give a straightforward analytic proof of the above mentioned result, motivated by a simple geometric idea, and then show that the general result is implied by its special case.

On the Skorokhod representation theorem

In this paper we present a variant of the well-known Skorokhod Representation Theorem. First we prove, given S S a Polish Space, that to a given continuous path α \alpha in the space of probability measures on S S , we can associate a continuous path in the space of S S -valued random variables on a nonatomic probability space (endowed with the topology of the convergence in probability). We call this associated path a lifting of α \alpha . An interesting feature of our result is that we can fix the endpoints of the lifting of α \alpha , as long as their distributions correspond to the respective endpoints of α \alpha . Finally, we also discuss and prove an n n -dimensional generalization of this result.