Loeb Measure Research Articles

Mixing and double recurrence in probability groups

We define a class of groups equipped with an invariant probability measure, which includes all compact groups and is closed under taking ultraproducts with the induced Loeb measure; in fact, this class also contains the ultraproducts of all locally compac

CONSTRUCTING NONSTANDARD HULLS AND LOEB MEASURES IN INTERNAL SET THEORIES

AbstractCurrently the two popular ways to practice Robinson’s nonstandard analysis are the model-theoretic approach and the axiomatic/syntactic approach. It is sometimes claimed that the internal axiomatic approach is unable to handle constructions relying on external sets. We show that internal frameworks provide successful accounts of nonstandard hulls and Loeb measures. The basic fact this work relies on is that the ultrapower of the standard universe by a standard ultrafilter is naturally isomorphic to a subuniverse of the internal universe.

A Real-Valued Measure on non-Archimedean Field Extensions of $$\mathbb{R}$$

We introduce a real-valued measure ${m_L}$ on non-Archimedean ordered fields $(\mathbb{F},<)$ that extend the field of real numbers $(\mathbb{R},<)$. The definition of ${m_L}$ is inspired by the Loeb measures of hyperreal fields in the framework of Robinson's analysis with infinitesimals. The real-valued measure ${m_L}$ turns out to be general enough to obtain a canonical measurable representative in $\mathbb{F}$ for every Lebesgue measurable subset of $\mathbb{R}$, moreover, the measure of the two sets is equal. In addition, $m_L$ it is more expressive than a class of non-Archimedean uniform measures. We focus on the properties of the real-valued measure in the case where $\mathbb{F}=\mathcal{R}$, the Levi-Civita field. In particular, we compare ${m_L}$ with the uniform non-Archimedean measure over $\mathcal{R}$ developed by Shamseddine and Berz, and we prove that the first is infinitesimally close to the second, whenever the latter is defined. We also define a real-valued integral for functions on the Levi-Civita field, and we prove that every real continuous function has an integrable representative in $\mathcal{R}$. Recall that this result is false for the current non-Archimedean integration over $\mathcal{R}$. The paper concludes with a discussion on the representation of the Dirac distribution by pointwise functions on non-Archimedean domains.

Loeb extension and Loeb equivalence

In [J. London Math. Soc. 69 (2004), pp. 258–272] Keisler and Sun leave open several questions regarding Loeb equivalence between internal probability spaces; specifically, whether under certain conditions, the Loeb measure construction applied to two such spaces gives rise to the same measure. We present answers to two of these questions, by giving two examples of probability spaces. Moreover, we reduce their third question to the following: Is the internal algebra generated by the union of two Loeb equivalent internal algebras a subset of their common Loeb extension? We also present a sufficient condition for a positive answer to this question.

On the converse law of large numbers

Given a triangular array with mn random variables in the nth row and a growth rate {kn}n=1∞ with lim sup n→∞(kn/mn)<1, if the empirical distributions converge for any subarrays with the same growth rate, then the triangular array is asymptotically independent. In other words, if the empirical distribution of any kn random variables in the nth row of the triangular array is asymptotically close in probability to the law of a randomly selected random variable among these kn random variables, then two randomly selected random variables from the nth row of the triangular array are asymptotically close to being independent. This provides a converse law of large numbers by deriving asymptotic independence from a sample stability condition. It follows that a triangular array of random variables is asymptotically independent if and only if the empirical distributions converge for any subarrays with a given asymptotic density in (0,1). Our proof is based on nonstandard analysis, a general method arisen from mathematical logic, and Loeb measure spaces in particular.

A monad measure space for logarithmic density

We provide a framework for proofs of structural theorems about sets with positive Banach logarithmic density. For example, we prove that if $A\subseteq \mathbb{N}$ has positive Banach logarithmic density, then $A$ contains an approximate geometric progression of any length. We also prove that if $A,B\subseteq \mathbb{N}$ have positive Banach logarithmic density, then there are arbitrarily long intervals whose gaps on $A\cdot B$ are multiplicatively bounded, a multiplicative version Jin's sumset theorem. The main technical tool is the use of a quotient of a Loeb measure space with respect to a multiplicative cut.

Extensions and Applications of the S-Measure Construction

AbstractS-measures are Loeb measures restricted to the sigma algebra generated by standard sets. This paper gives new extensions of the S-measure machinery, with applications to standard measure theory.

The Boolean prime ideal theorem and products of cofinite topologies

We show: The Boolean Prime Ideal theorem is equivalent to each one of the statements: “For every family of compact spaces, for every family of basic closed sets of the product with the fip there is a family of subbasic closed sets () with the fip such that for every ”. “For every compact Loeb space (the family of all non empty closed subsets of has a choice function) and for every set X the product is compact”. (: the axiom of choice restricted to families of finite sets) implies “every well ordered product of cofinite topologies is compact” and “every well ordered basic open cover of a product of cofinite topologies has a finite subcover”. (: the axiom of choice restricted to countable families of finite sets) iff “every countable product of cofinite topologies is compact”. (: every filter of extends to an ultrafilter) is equivalent to the proposition “for every compact Loeb space having a base of size and for every set X of size the product is compact”.

A convex structure on sofic embeddings

AbstractNathanial Brown [Topological dynamical systems associated to ${\mathit{II}}_{1} $-factors. Adv. Math. 227(4), 1665–1699] introduced a convex-like structure on the set of unitary equivalence classes of unital *-homomorphisms of a separable type ${\mathit{II}}_{1} $ factor into ${R}^{\omega } $ (ultrapower of the hyperfinite factor). The goal of this paper is to introduce such a structure on the set of sofic representations of groups. We prove that if the commutant of a representation acts ergodically on the Loeb measure space then that representation is an extreme point.

Set-valued functions, Lebesgue extensions and saturated probability spaces

Recent advances in the theory of distributions of set-valued functions have been shaped by counterexamples which hinge on the non-existence of measurable selections with requisite properties. These examples, all based on the Lebesgue interval, and initially circumvented by Sun in the context of Loeb spaces, have now led Keisler and Sun (KS) to establish a comprehensive theory of the distributions of set-valued functions on saturated probability spaces (introduced by Hoover and Keisler). In contrast, we show that a countably-generated extension of the Lebesgue interval suffices for an explicit resolution of these examples; and furthermore, that it does not contradict the KS necessity results. We draw the fuller implications of our theorems for integration of set-valued functions, for Lyapunov's result on the range of vector measures and for the theory of large non-anonymous games.

Simple nonstandard proofs of Daniell–Kolmogorov-type theorems

This brief note employs a proof technique essentially due to Lindstrom, based on the Loeb measure construction, to obtain short and simple proofs of theorems of the Daniell–Kolmogorov type.

Young measures as probability distributions of Loeb spaces

First we give a simple proof for a folklore result, which we call the Fundamental Theorem of Young Measures in a general framework. In the second part, we deal with the explicit representation of Young measures on Euclidean spaces. Young measure is an abstract concept in the sense that when it is described, it needs all continuous functions ϕ ( y ) \phi (y) and all L 1 L^1 functions f ( x ) f(x) in the realm of standard analysis. However, we found that in the framework of nonstandard analysis, Young measures at almost all points are proved to be probability distributions for some random variables on some Loeb spaces defined in the monads of those points. This means that we can describe this Young measure without using f ( x ) f(x) and ϕ ( y ) \phi (y) . This also leads to the concrete computation of Young measures.

Radon-Nikodym theorem in signed Loeb space

This paper gives the Radon-Nikodym theorem in signed Loeb space under ℵ1-saturated nonstandard model. First, the nonstandard characterization of absolute continuity is discussed, on which Radon-Nikodym theorem in signed Loeb space is obtained. Then, some facts about a finite signed Loeb measure and its variation are shown.

Representing measures in potential theory and an ideal boundary

A primary motivating application of the author’s work on measure theory was a nonstandard construction of standard representing measures for positive harmonic functions. That work yielded new standard weak convergence methods for constructing such measures on spaces of extreme harmonic functions in very general settings. The search for a Martin-type ideal boundary for the placement of those measures resulted in a new almost everywhere regular boundary that supported the representing measures for a large proper subclass of all nonnegative harmonic functions. In this note, we outline the construction of the rich measure spaces that are now called Loeb measure spaces in the literature. We then review the application of these measure spaces to the construction of representing measures. We finish with the problem of constructing an appropriate boundary associated with the nonstandard construction of general representing measures that supports all of those measures.

Pushing down infinite Loeb measures

Sufficient conditions are given under which the standard part map on an arbitrary Hausdorff space can be used to push down an infinite nonstandard measure. This makes it easier to construct standard infinite Borel measures using nonstandard techniques.

Purification and saturation

This paper illustrates the general technique established in 1984 by Hoover and Keisler for extending certain types of results from atomless Loeb measure spaces to measure spaces that we shall call countably generated. The Hoover-Keisler technique is applied here to further extend the authors' 2006 generalization of a theorem of Dvoretzky, Wald and Wolfowitz on the purification of measure-valued maps. The authors' 2006 result was first extended to these more general spaces by K. Podczeck in 2007; he used new results in functional analysis produced for that purpose. This paper demonstrates that, in general, such extensions follow from the Hoover-Keisler technique. Moreover, adaptations of counterexamples from earlier papers show that the extension obtained here holds only for nowhere countably generated spaces.

On the existence of pure-strategy equilibria in large games

Over the years, several formalizations and existence results for games with a continuum of players have been given. These include those of Schmeidler [D. Schmeidler, Equilibrium points of nonatomic games, J. Stat. Phys. 4 (1973) 295–300], Rashid [S. Rashid, Equilibrium points of non-atomic games: Asymptotic results, Econ. Letters 12 (1983) 7–10], Mas-Colell [A. Mas-Colell, On a theorem by Schmeidler, J. Math. Econ. 13 (1984) 201–206], Khan and Sun [M. Khan, Y. Sun, Non-cooperative games on hyperfinite Loeb spaces, J. Math. Econ. 31 (1999) 455–492] and Podczeck [K. Podczeck, On purification of measure-valued maps, Econ. Theory 38 (2009) 399–418]. The level of generality of each of these existence results is typically regarded as a criterion to evaluate how appropriate is the corresponding formalization of large games. In contrast, we argue that such evaluation is pointless. In fact, we show that, in a precise sense, all the above existence results are equivalent. Thus, all of them are equally strong and therefore cannot rank the different formalizations of large games.

Loeb measure, finitely additive measures, and a theorem of Banach

The Loeb measure construction from nonstandard analysis is applied to two theorems in standard measure theory. In both cases the essential simplification offered by the approach is the ability to work with a σ-additive measure space, even if the hypotheses only guarantee finite additivity. The key to this simplification is the property of \({\aleph_1}\)-saturated nonstandard models, that any finitely additive measure on an internal algebra extends immediately to a σ-additive measure.

On purification of measure-valued maps

This paper presents new methods to obtain purification results for continuum games, which don’t make use of the “many more players than strategies” assumption (Yannelis in Econ Theory (in press) 2007) or of Loeb spaces (Loeb and Sun in Illinois J Math 50, 747–762, 2006). The approach presented doesn’t use nonstandard analysis; it is based on standard measure theory and in particular on the super-nonatomicity notion introduced in Podczeck (J Math Econ (in press) 2007).

Saturation and the integration of Banach valued correspondences

This note illustrates that the saturation property of a probability space can be used to routinely generalize results on the integration of Banach valued correspondences over a Loeb measure space to those over an arbitrary saturated probability space. On the other hand, the saturation property is also necessary for the validity of those results when the target space is infinite dimensional.