Nonstandard Measure Theory Research Articles

The arithmetic of cuts in models of arithmetic

We present a number of results on the structure of initial segments of models of Peano arithmetic with the arithmetic operations of addition, subtraction, multiplication, division, exponentiation and logarithm. Each of the binary operations introduced is defined in two dual ways, often with quite different results, and we attempt to systematise the issues and show how various calculations may be carried out. To understand the behaviour of addition and subtraction we introduce a notion of derivative on cuts, analogous to differentiation in the calculus. Multiplication, division and other operations are described by higher order versions of derivative. The work here is presented as important preliminary work related to a nonstandard measure theory of non‐definable bounded subsets of a model of Peano arithmetic.

Compatibilism can resist prepunishment: a reply to Smilansky

Bernstein, A. R. and F. Wattenberg. 1969. Non-standard measure theory. In Appli cations of Model Theory of Algebra, Analysis, and Probability, ed. W. A. J. Luxemburg. New York: Holt, Rinehard and Winston. Lewis, D. 1980. A subjectivist's guide to objective probability. In Ifs: Conditionals, Belief, Decision, Chance and Time, ed. W. L. Harper, R. Stalnaker and G. Pearce, 1981: 267-97. Boston: Reidel. Williamson, T. 2007. How probable is an infinite sequence of heads? Analysis 67: 173-80.

Formula omitted]-homomorphisms from the Borel [formula omitted]-algebra into the Loeb [formula omitted]-algebra

In nonstandard measure theory the standard part map is very useful to represent standard measures by Loeb measures. We give here a different method of representing measures using the concept of a σ-homomorphism. As an application a measure extension theorem is derived. Finally a nonstandard proof of a result of Pták is given showing that his assumption of nonmeasurable cardinality is redundant.

Developments in Constructive Nonstandard Analysis

AbstractWe develop a constructive version of nonstandard analysis, extending Bishop's constructive analysis with infinitesimal methods. A full transfer principle and a strong idealisation principle are obtained by using a sheaf-theoretic construction due to I. Moerdijk. The construction is, in a precise sense, a reduced power with variable filter structure. We avoid the nonconstructive standard part map by the use of nonstandard hulls. This leads to an infinitesimal analysis which includes nonconstructive theorems such as the Heine–Borel theorem, the Cauchy–Peano existence theorem for ordinary differential equations and the exact intermediate-value theorem, while it at the same time provides constructive results for concrete statements. A nonstandard measure theory which is considerably simpler than that of Bishop and Cheng is developed within this context.

Lifting Theorems in Nonstandard Measure Theory

1. A nonstandard capacity construction, analogous to Loeb’s measure construction, is developed. Using this construction and Choquet’s Capacitability theorem, it is proved that a Loeb measurable function into a general (not necessarily second countable) space has a lifting precisely when its graph is ’almost’ analytic. This characterization is used to generalize and simplify some known lifting existence theorems. 2. The standard notion of ’Lusin measurability’ is related to the nonstandard notion of admitting a ’two-legged’ lifting. An immediate consequence is a new and simple proof of the general Lusin theorem. Another consequence is the existence of a Loeb measurable function, not admitting a lifting, into a relatively small topological space.

Lifting theorems in nonstandard measure theory

1. A nonstandard capacity construction, analogous to Loeb’s measure construction, is developed. Using this construction and Choquet’s Capacitability theorem, it is proved that a Loeb measurable function into a general (not necessarily second countable) space has a lifting precisely when its graph is ’almost’ analytic. This characterization is used to generalize and simplify some known lifting existence theorems. 2. The standard notion of ’Lusin measurability’ is related to the nonstandard notion of admitting a ’two-legged’ lifting. An immediate consequence is a new and simple proof of the general Lusin theorem. Another consequence is the existence of a Loeb measurable function, not admitting a lifting, into a relatively small topological space.

Nigel J. Cutland. Nonstandard measure theory and its applications. The bulletin of the London Mathematical Society, vol. 15 (1983), pp. 529–589.

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Nonstandard Measure Theory and its Applications

Mathematiques non standard. Theorie de la mesure non standard. Mesures de Lœb. Representations des mesures par des mesures de Lœb. Construction de mesures. Probabilite et processus stochastiques. Analyse stochastique

A simple proof of the ergodic theorem using nonstandard analysis

A simple proof of the individual ergodic theorem is given. The essential tool is the nonstandard measure theory developed by P. Loeb. Any dynamical system on an abstract Lebesgue space can be represented as a factor of a “cyclic” system with a hyperfinite cycle. The ergodic theorem for such a “cyclic” system is almost trivial because of its simple structure. The general case follows from this special case.

Nonstandard set Theory

Infinitely small and infinitely large quantities were systematically introduced into mathematics with the invention of calculus by Newton and Leibniz. The use of such quantities, however, was accompanied by logical contradictions, which mathematicians of the seventeenth and eighteenth centuries were unable to resolve. Although the method of infinitesimals generally yielded correct results, no one ever succeeded in formulating a precise, noncontradictory set of rules governing these objects; and infinitesimal quantities were gradually displaced (at least, in pure mathematics) by the familiar e-d calculus. A mathematically sound model of infinitely small and infinitely large objects became possible only after advances in mathematical logic in the twentieth century. Nonstandard Analysis, developed by A. Robinson in 1960, not only provided foundations for the calculus of infinitesimals in the classical spirit but also enabled mathematicians to use objects in ways that could not be attempted on the basis of vague, intuitive understanding alone. Since then, interesting applications were found in various branches of mathematics, mathematical physics, and economics. Robinson's exposition in [10] and its subsequent simplifications unfortunately involve the cumbersome apparatus of mathematical logic. Our aim here is to present methods of Nonstandard Analysis at a level of formalism customary in other branches of mathematics. We view nonstandard objects as ideal, imaginary elements adjoined to the universe of the standard mathematics and formulate a few simple and reasonably intuitive principles governing their behavior. We then show, on examples selected to illustrate a variety of nonstandard constructions, how nonstandard mathematics can be developed from these principles. The basic framework for Nonstandard Analysis is presented in ?? 1-3; this system was introduced in [3], where its relative consistency with respect to the Zermelo-Fraenkel set theory is shown. We examine the real line and some concepts of general topology from our point of view in ??4-5; these results can be found in Robinson [10] and Luxemburg [7], [8]. Section 6 is devoted to nonstandard measure theory; our approach is basically that of Loeb [6] (except that we construct Loeb's extension in Theorem 3 of ?6 directly, rather than using Caratheodory's Theorem), with some ideas coming from Anderson [1]. The final ?7, part I, contains a more formal description of the logical foundations, and parts II and III discuss the relationship between our approach and the classical one based on higher-order nonstandard models, as well as some other axiomatizations of Nonstandard Analysis.

Nonstandard measure theory: avoiding pathological sets

The main results in this paper concern representing Lebesgue measure by nonstandard measures which avoid certain pathological sets. An (external) set E is S-thin if InfmA|A standard,* A ⊇ E A\, \supseteq \,E = 0 and Q-thin if Inf*mA|A internal, A ⊇ E A\, \supseteq \,E = 0. It is shown that any *finite sample which represents Lebesgue measure avoids every S-thin set and that given any Q-thin set E there is a *finite sample avoiding E which represents Lebesgue measure. In the last part of the paper a particular pathological set H ⊆ ∗ [ 0 , 1 ] {\mathcal {H}}\,\, \subseteq \, * \left [ {0,\,1} \right ] is constructed which is important for the study of approximate limits, derivatives etc. It is shown that every *finite sample which represents Lebesgue measure assigns inner measure zero and outer measure one to this set and that Loeb measure does the same. Finally, it is shown that Loeb measure can be extended to a σ \sigma -algebra including H {\mathcal {H}} in such a way that H {\mathcal {H}} is assigned zero measure.

Nonstandard measure theory–Hausdorff measure

In this paper it is shown that the Hausdorff measures λ t {\lambda ^t} for t ∈ [ 0 , ∞ ) t \in [0,\infty ) can be simultaneously represented as ∗ ^\ast finite counting measures in an appropriate nonstandard model. That is, the following theorem is proved. Theorem. Suppose X is a metric space and ν \nu is an infinite positive ∗ ^\ast integer. Then there is a ∗ ^\ast finite set G such that for every standard t ∈ [ 0 , ∞ ) t \in [0,\infty ) and every λ t {\lambda ^t} -integrable Borel function, f : X → R f:X \to {\mathbf {R}} , \[ ∫ f d λ t = St ( 1 ν t ∑ x ∈ G ∗ f ( x ) ) . \int {f\;d{\lambda ^t}} = {\text {St}}\left ( {\frac {1}{{{\nu ^t}}}\sum \limits _{x \in G} {*f(x)} } \right ). \]