Nonatomic Probability Measures Research Articles

Conditional probability and probability updating

The conditional probability formula is supposed to reflect the correct updating of probability assignments when new information is incorporated. Starting from a non-atomic probability measure, it is proved that the conditional probability formula provides the only transformed probability measure satisfying a “minimum requirement” relational assumption. This result applies to the standard Bayesian parametric model.

Simple fair division of a square

Suppose we are given a cake represented by the unit interval to be divided among agents evaluating the pieces of the cake by nonatomic probability measures. It is known that we can divide the unit interval into contiguous and connected pieces and assign them to the agents in such a way that the values of the pieces are equal according to the individual agents measures. Such division is said to be equitable and simple. In this paper we show that an equitable and simple division also exists in the case of dividing two-dimensional cake represented by the unit square. In this case, by simple division we mean dividing the unit square firstly by horizontal cuts, and then partition the resulting rectangles by vertical cuts. We give a method of obtaining a proportional and simple division of this cake. Furthermore, we prove the existence of proportional, equitable and simple division.

How to obtain an equitable optimal fair division

A nonlinear programming method is used for finding an equitable optimal fair division of the unit interval [0, 1) among n players. Players’ preferences are described by nonatomic probability measures mu _{1},ldots ,mu _{n} with density functions having piecewise strict monotone likelihood ratio property. The presented algorithm can be used to obtain also an equitable varepsilon -optimal fair division in case of measures with arbitrary differentiable density functions. An example of an equitable optimal fair division for three players is given.

Invariance of nonatomic measures on effect algebras

Abstract The present paper deals with invariance of nonatomic measures defined on effect algebras. Firstly, it is proved that if μ is a nonatomic and continuous probability measure defined on a σ-complete effect algebra L, then it satisfies para-Darboux property. Then, the invariance between continuous probability measures m and μ defined on a σ-complete effect algebra L is established when μ is nonatomic satisfying para-Darboux property on L.

The range of a nonatomic vector measure in ℝ3

Using the Neyman-Pearson lemma we give a method for finding the range of three nonatomic probability measures. Next we use this result to construct an explicit form of this range in the case, where the densities corresponding to these measures satisfy an additional assumption.

Optimal Fair Division for Measures with Piecewise Linear Density Functions

A nonlinear programming method is used for finding an optimal fair division of the unit interval [Formula: see text] among [Formula: see text] players. Preferences of players are described by nonatomic probability measures [Formula: see text] with piecewise linear (PWL) density functions. The presented algorithm can be applied for obtaining “almost” optimal fair divisions for measures with arbitrary density functions approximable by PWL functions. The number of cuts needed for obtaining such divisions is given.

Some Condition for Scalar and Vector Measure Games to Be Lipschitz

We provide a characterization for vector measure gamesν=f∘μinpNA∞, withμvector of nonatomic probability measures, analogous to the one of Tauman for games inpNA, and also provide a necessary and sufficient condition for a particular class of vector measure games to belong toAC∞.

Optimal weak type estimates for dyadic-like maximal operators

We provide sharp weak estimates for the distribution function of Mwhen onwe impose L 1 , L q and L p;1 restrictions. Here M is the dyadic maximal operator associated to a tree T on a non-atomic probability measure space. As a consequence we produce that the inequality jjMTjjp;1 • jjjjjjp;1 is sharp allowing every possible value for the L 1 and the L q norm for a fixed q such that 1 < q < p, where jjj ¢ jjjp;1 is the integral norm on and jj ¢ jjp;1 the usual quasi norm on L p;1 .

On the existence of equitable cake divisions

Let the cake be represented by the unit interval and let each player have a valuation expressed by a nonatomic probability measure. A cake division is said to be equitable if the value of the piece assigned to a player by his measure is the same for all players. We show that for any number n of players in any order an equitable division exists, giving each player a contiguous cake piece. Moreover, there is at least one order in which the common value is not smaller than 1/n.

The speed with which an orbit approaches a limit point

If is a continuous map of a compact metric space , and if is a sequence of positive reals converging to 0, we investigate the properties of the set . We show that is a dense subset of for every when x is a recurrent point, even though can be disjoint with the orbit of x for some . Under the assumption that f has an invariant non-atomic Borel probability measure , we prove results to the effect that (i) there is a uniform upper limit to the speed with which the orbit of each x can approach y for -almost every , (ii) if is ergodic with full support and if is the set of points having dense orbits, then for -almost every and for every there is a uniform upper limit to the speed with which the orbit of x can approach y. Next, using as a useful tool in proofs, we establish the following. If f is totally transitive and X is infinite, then there is a dense subset which is a countable union of Cantor sets such that and for any two distinct and any two distinct . If f is a transitive map enjoying a certain type of continuity in the backward direction, then f has a residual set of points with dense backward orbits.

On the computability of equitable divisions

Let the cake be represented by the unit interval of reals, with players having private valuations expressed by nonatomic probability measures. The aim is to find a cake division which assigns to each of n players one contiguous piece (a simple division) in such a way that the value each player receives (by her own measure) is the same for all players and this common value is at least 1/n. It is known that such divisions always exist, however, we show that there is no finite algorithm to find them already for three players. Therefore we propose an algorithm that for any given ε>0 finds, in a finite number of steps, a simple division such that the values assigned to players differ by at most ε>0.

How to obtain a range of a nonatomic vector measure in [formula omitted

Using the Neyman–Pearson lemma we give a method for effecting a range of two nonatomic probability measures. We use this result to find an optimal partition in the cake division problem.

A new class of convex games on σ-algebras and the optimal partitioning of measurable spaces

We introduce μ-convexity, a new kind of game convexity defined on a σ-algebra of a nonatomic finite measure space. We show that μ-convex games are μ-average monotone. Moreover, we show that μ-average monotone games are totally balanced and their core contains a nonatomic finite signed measure. We apply the results to the problem of partitioning a measurable space among a finite number of individuals. For this problem, we extend some results known for the case of individuals’ preferences that are representable by nonatomic probability measures to the more general case of nonadditive representations.

Notes on the nuclearity of random subalgebras of $${\mathcal {O}_2}$$

There exists a separable exact C*-algebra A which contains all separable exact C*-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x 1, x 2, ... we have $${\rm {lim}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0$$ . Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra $${\mathcal {O}_2}$$ such that for an independent sequence x 1, x 2, ... of μ-distributed random elements x i we have $${\rm {lim\, inf}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0$$ . We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C([0, 1] d ) is d.

On hyperbolic measures and periodic orbits

We prove that if a diffeomorphism on a compact manifold preserves a nonatomic ergodic hyperbolic Borel probability measure, then there exists a hyperbolic periodic point such that the closure of its unstable manifold has positive measure. Moreover, the support of the measure is contained in the closure of all such hyperbolic periodic points. We also show that if an ergodic hyperbolic probability measure does not locally maximize entropy in the space of invariant ergodic hyperbolic measures, then there exist hyperbolic periodic points that satisfy a multiplicative asymptotic growth and are uniformly distributed with respect to this measure.

CHOQUET INSURANCE PRICING: A CAVEAT

We show that, if prices in a market are Choquet expectations, the existence of one frictionless asset may force the whole market to be frictionless. Any risky asset will cause this collapse if prices depend only on the distribution with respect to a given nonatomic probability measure; the frictionless asset has to be fully revealing if such dependence is not assumed. Similar considerations apply to law‐invariant coherent risk measures.

How to cut a pizza fairly: Fair division with decreasing marginal evaluations

Existential and constructive solutions to the classic problems of fair division are known for individuals with constant marginal evaluations. By considering nonatomic concave capacities instead of nonatomic probability measures, we extend some of these results to the case of individuals with decreasing marginal evaluations.

Values of non-atomic vector measure games

There is a value (of norm one) on the closed space of games that is generated by all games of bounded variationf o μ, whereμ is a vector of non-atomic probability measures andf is continuous at 0=μ(o) and atμ(I).

On the Geometry of Cake Division

We study partitions of a “cake” C among n players. Each player uses a countably additive non-atomic probability measure to evaluate the sizes of pieces of cake. If the players' measures are m1,m2,…,mn, then the “Individual Pieces Set,” which we studied before (2000, J. Math. Econom.33, 401–424), is the set {(m1(P1),m2(P2),…,mn(Pn)):〈P1,P2,…,Pn〉 is a partition of C}. We continue our study of this set here. Our motivating question is: What are the possible shapes of such sets? We give an exact characterization for n=2, establish some partial results for n=3, and close with open questions.

On selection theorems with decomposable values

The main result of the paper asserts that for every separable measurable space $(T,\mathfrak F,\mu)$, where $\mathfrak F$ is the $\sigma$-algebra of measurable subsets of $T$ and $\mu$ is a nonatomic probability measure on $\mathfrak F$, every Banach space $E$ and every paracompact space $X$, each dispersible closed-valued mapping $F: x \rightsquigarrow L_1(T,E)$ of $X$ into the Banach space $L_1(T,E)$ of all Bochner integrable functions $u: T\to E$, admits a continuous selection. Our work generalizes some results of Gon\v carov and Tol'stonogov.