Lebesgue Integration Theory Research Articles

Logarithms, constructible functions and integration on non-archimedean models of the theory of the real field with restricted analytic functions with value group of finite archimedean rank

Given a model of the theory of the real field with restricted analytic functions such that its value group has finite archimedean rank we show how one can extend the restricted logarithm to a global logarithm with values in the polynomial ring over the model with dimension the archimedean rank. The logarithms are determined by algebraic data from the model, namely by a section of the model and by an embedding of the value group into its Hahn group. If the archimedean rank of the value group coincides with the rational rank the logarithms are equivalent. We illustrate how one can embed such a logarithm into a model of the real field with restricted analytic functions and exponentiation. This allows us to define constructible functions with good lifting properties. As an application we establish a Lebesgue measure and integration theory with values in the polynomial ring, extending and strengthening the construction in [T. Kaiser: Lebesgue measure and integration theory on non-archimedean real closed fields with archimedean value group. Proc. Lond. Math. Soc. 116 (2018), no. 2, 209-247.].

Algebraic Theories and Commutativity in a Sheaf Topos

For any site of definition $$\mathcal {C}$$ C of a Grothendieck topos $$\mathcal {E}$$ E, we define a notion of a $$\mathcal {C}$$ C-ary Lawvere theory $$\tau : \mathscr {C} \rightarrow \mathscr {T}$$ τ:C→T whose category of models is a stack over $$\mathcal {E}$$ E. Our definitions coincide with Lawvere’s finitary theories when $$\mathcal {C}=\aleph _0$$ C=ℵ0 and $$\mathcal {E} = {{\,\mathrm{\mathbf {Set}}\,}}$$ E=Set. We construct a fibered category $${{\,\mathrm{\mathbf {Mod}}\,}}^{\mathscr {T}}$$ ModT of models as a stack over $$\mathcal {E}$$ E and prove that it is $$\mathcal {E}$$ E-complete and $$\mathcal {E}$$ E-cocomplete. We show that there is a free-forget adjunction $$F \dashv U: {{\,\mathrm{\mathbf {Mod}}\,}}^{\mathscr {T}} \leftrightarrows \mathscr {E}$$ F⊣U:ModT⇆E. If $$\tau $$ τ is a commutative theory in a certain sense, then we obtain a “locally monoidal closed” structure on the category of models, which enhances the free-forget adjunction to an adjunction of symmetric monoidal $$\mathcal {E}$$ E-categories. Our results give a general recipe for constructing a monoidal $$\mathcal {E}$$ E-cosmos in which one can do enriched $$\mathcal {E}$$ E-category theory. As an application, we describe a convenient category of linear spaces generated by the theory of Lebesgue integration.

The Vitali convergence in measure theorem of nonlinear integrals

The Vitali convergence in measure theorem for the abstract Lebesgue integral is fundamental in Lebesgue integration theory and yields the bounded convergence theorem and the dominated convergence theorem as its applications. In this paper, in a unified way using the perturbation method the Vitali convergence in measure theorem is established for nonlinear integrals such as the Choquet, Šipoš, Sugeno, and Shilkret integrals, and their symmetric and asymmetric extensions. It is derived from the Fatou and the reverse Fatou type lemmas for perturbative nonlinear integral functionals.

Fubini’s Theorem for Non-Negative or Non-Positive Functions

Summary The goal of this article is to show Fubini’s theorem for non-negative or non-positive measurable functions [10], [2], [3], using the Mizar system [1], [9]. We formalized Fubini’s theorem in our previous article [5], but in that case we showed the Fubini’s theorem for measurable sets and it was not enough as the integral does not appear explicitly. On the other hand, the theorems obtained in this paper are more general and it can be easily extended to a general integrable function. Furthermore, it also can be easy to extend to functional space Lp [12]. It should be mentioned also that Hölzl and Heller [11] have introduced the Lebesgue integration theory in Isabelle/HOL and have proved Fubini’s theorem there.

Lebesgue measure and integration theory on non-archimedean real closed fields with archimedean value group

Given a non-archimedean real closed field with archimedean value group which contains the reals, we establish for the category of semialgebraic sets and functions a full Lebesgue measure and integration theory such that the main results from the classical setting hold. The construction involves methods from model theory, o-minimal geometry and valuation theory. We set up the construction in such a way that it is determined by a section of the valuation. If the value group is isomorphic to the group of rational numbers the construction is uniquely determined up to isomorphism. The range of the measure and integration is obtained in a controled and tame way from the real closed field we start with. The main example is given by the case of the field of Puiseux series where the range is the polynomial ring in one variable over this field.

On some Diophantine fourier series

We continue our study on arithmetical Fourier series by considering two Fourier series which are related to Diophantine analysis. The first one was studied by Hardy and Littlewood in connection with the classification of numbers and the second one was studied by Hartman and Wintner by Lebesgue integration theory.

Integral of the Clarke subdifferential mapping and a generalized Newton–Leibniz formula

In the theory of Lebesgue integration it has been proved that if f is a real Lipschitz function defined on a segment [ a , b ] ⊂ R , then the Newton–Leibniz formula ∫ a b f ′ ( x ) d x = f ( b ) − f ( a ) (the fundamental theorem of calculus) holds. This paper extends the fact to the case where the Fréchet derivative f ′ ( ⋅ ) (which is defined almost everywhere on [ a , b ] by the Rademacher theorem) and the Lebesgue integral are replaced, respectively, by the Clarke subdifferential mapping ∂ C f ( ⋅ ) and the Aumann (set-valued) integral. Among other things, we show that f ( b ) − f ( a ) ∈ ∫ a b ∂ C f ( x ) d x and the equality ∫ a b ∂ C f ( x ) d x = { f ( b ) − f ( a ) } is valid if and only if f is strictly Hadamard differentiable almost everywhere on [ a , b ] . The result is derived from a general representation formula, which we obtain herein for the integral of the Clarke subdifferential mapping of a Lipschitz function defined on a separable Banach space.

Computing the Bayes Factor from a Markov Chain Monte Carlo Simulation of the Posterior Distribution

Determining the marginal likelihood from a simulated posterior distribution is central to Bayesian model selection but is computationally challenging. The often-used harmonic mean approximation (HMA) makes no prior assumptions about the character of the distribution but tends to be inconsistent. The Laplace approximation is stable but makes strong, and often inappropriate, assumptions about the shape of the posterior distribution. Here, I argue that the marginal likelihood can be reliably computed from a posterior sample using Lebesgue integration theory in one of two ways: 1) when the HMA integral exists, compute the measure function numerically and analyze the resulting quadrature to control error; 2) compute the measure function numerically for the marginal likelihood integral itself using a space-partitioning tree, followed by quadrature. The first algorithm automatically eliminates the part of the sample that contributes large truncation error in the HMA. Moreover, it provides a simple graphical test for the existence of the HMA integral. The second algorithm uses the posterior sample to assign probability to a partition of the sample space and performs the marginal likelihood integral directly. It uses the posterior sample to discover and tessellate the subset of the sample space that was explored and uses quantiles to compute a representative field value. When integrating directly, this space may be trimmed to remove regions with low probability density and thereby improve accuracy. This second algorithm is consistent for all proper distributions. Error analysis provides some diagnostics on the numerical condition of the results in both cases.

Rethinking the Elementary Real Analysis Course

Finally, the reader will probably observe the conspicuous absence of the time-honored topic in calculus courses, the “Riemann integral.” It may well be suspected that, had it not been for its prestigious name, this would have been dropped long ago, for (with due reverence to Riemann’s genius) it is certainly quite clear to any working mathematician that nowadays such a “theory” has at best the importance of a mildly interesting exercise in the general theory of measure and integration. Only the stubborn conservatism of academic tradition could freeze it into a regular part of the curriculum, long after it had outlived its historical importance. Of course, it is perfectly feasible to limit the integration process to a category of functions which is large enough for all purposes of elementary analysis, but close enough to the continuous functions to dispense with any consideration drawn from measure theory; this is what we have done by defining only the integral of regulated functions. When one needs a more powerful tool there is no point in stopping halfway, and the general theory of (Lebesgue) integration is the only sensible answer.

Feynman's Path Integrals and Henstock's Non-Absolute Integration

R. Feynman formulated quantum mechanics in terms of integrals over spaces of paths (Feynman path integrals). But the absolute value of Feynman's integrand is not integrable. And his integrand does not generate a measure. So Lebesgue integration theory could not be used by Feynman. To establish the equivalence of his theory with the traditional formulation of quantum mechanics, Feynman gave an argument that his path integral satisfies Schrödinger's equation. This paper gives a proof of this part of Feynman's theory. To justify Feynman's and other investigators' use of the language and concepts of integration and probability theory, and to justify taking limits under the integral sign in Feynman's integral, we use R. Henstock's approach to non-absolute integration, which does not require the measure concept, and for which the absolute value of the integrand need not be integrable.

A Riemann Sum Upper Bound in the Riemann--Lebesgue Theorem

The Riemann--Lebesgue theorem is commonly proved in a few strokes using the theory of Lebesgue integration. Here, the upper bound $2\pi|c_k(f)|\le S_k(f)-s_k(f)$ for the Fourier coefficients ck is proved in terms of majoring and minoring Riemann sums Sk(f) and sf(k), respectively, for Riemann integrable functions f(x). This proof has been used in a course on methods of applied mathematics.

A multidimensional analogue of the Denjoy-Perron-Henstock-Kurzweil integral

Several new multidimensional integration theories that generalize or are analogous to the Denjoy-Perron-Henstock-Kurzweil integral were developed in recent years by several authors (e.g. [H], [L] (Generalized Riemann integral), [CD] (Generalized Denjoy integral), [M1] (GP-integral), [KMP] (BV-integral)). The purpose in the development of these theories has been to examine more general versions of the classical theorems in Lebesgue integration theory, such as the Divergence theorem, Fubini’s theorem, or convergence theorems. In this paper, we propose a simple elementary multidimensional integration that has a number of advantages also. First, as remarked in [L], unlike the onedimensional case, a drawback of the known multidimensional integrals is that one cannot develop in the same system both Divergence and Fubini type theorems. This can be done with the integral presented here. Second, one main goal of the above theories has been to weaken the smoothness condition on the vector fields in the Divergence theorem. The continuous differentiability of the vector fields was replaced by their continuity and their pointwise, or asymptotic, or a.e. differentiability (with some other supplementary conditions). Here, we can remove all hypotheses about differentiability and prove a Divergence theorem for the class of all continuous vector fields (in fact, for a larger class of distributions). The third point concerns convergence theorems. In some of the previous integration theories, the convergence theorems are rather complicated (see e.g. Ch. 5, [L]), and in some others, they seem

Possibility and necessity integrals

In this paper, we introduce seminormed and semiconormed fuzzy integrals associated with confidence measures. These confidence measures have a field of sets as their domain, and a complete lattice as their codomain. In introducing these integrals, the analogy with the classical introduction of Lebesgue integrals is explored and exploited. It is amongst other things shown that our integrals are the most general integrals that satisfy a number of natural basic properties. In this way, our dual classes of fuzzy integrals constitute a significant generalization of Sugeno's fuzzy integrals. A large number of important general properties of these integrals is studied. Furthermore, and most importantly, the combination of seminormed fuzzy integrals and possibility measures on the one hand, and semiconormed fuzzy integrals and necessity measures on the other hand, is extensively studied. It is shown that these combinations are very natural, and have properties which are analogous to the combination of Lebesgue integrals and classical measures. Using these results, the very basis is laid for a unifying measure- and integral-theoretic account of possibility and necessity theory, in very much the same way as the theory of Lebesgue integration provides a proper framework for a unifying and formal account of probability theory.

An elementary proof of an equivalence theorem relevant in the theory of optimization

The authors give an elementary proof of an equivalence theorem of analysis which is often used in optimization theory. The theorem asserts that certain conditions are equivalent to weak convergence inL1. One is the Dunford-Pettis condition concerning absolute integrability. Two others are expressed in terms of Nagumo functions, and can be thought of as growth properties. The original proofs of the various parts of the theorem are scattered in different and specialized mathematical publications. The authors feel it useful to present here a straightforward proof of the various parts in terms of standard Lebesgue integration theory.

An elementary proof of an equivalence theorem relevant in the theory of optimization

The authors give an elementary proof of an equivalence theorem of analysis which is often used in optimization theory. The theorem asserts that certain conditions are equivalent to weak convergence inL 1. One is the Dunford-Pettis condition concerning absolute integrability. Two others are expressed in terms of Nagumo functions, and can be thought of as growth properties. The original proofs of the various parts of the theorem are scattered in different and specialized mathematical publications. The authors feel it useful to present here a straightforward proof of the various parts in terms of standard Lebesgue integration theory.

Continuum calculus. II. The heterogeneous continuous functional differentiation applied to the Feynman path integral

The continuum calculus proposed previously [L. L. Lee, J. Math. Phys. 17, 1988 (1976)] is here extended to the study of continuous differentiation of a functional. The result, called HCFD, is shown to be the inverse operation of the functional integration for Feynman path integrals, in analogy to the case in ordinary differential calculus. The class of ’’separable’’ functionals is defined, which are useful in the derivation of the theory, playing a role similar to that of the characteristic functions in the Lebesgue theory of integration. A Radon–Nikodym type derivative is introduced in the definition of the continuous derivative for a general Banach algebra. This development constitutes a functional calculus of the continuum type. Comparisons with other types of functional derivatives are also made.

The Structure of Lebesgue Integration Theory. By G. Temple. Pp. 184. £3·75. 1971. (Oxford University Press.)

The Structure of Lebesgue Integration Theory. By G. Temple. Pp. 184. £3·75. 1971. (Oxford University Press.)

A Mathematical Idea: Lebesgue's Theory of Integration . Its Origins and Development. Thomas Hawkins. University of Wisconsin Press, Madison, 1970. xvi, 228 pp., illus. $12.50.; Classical and Modern Integration Theories . Ivan N. Pesin. Translated from the Russian edition (Moscow, 1966) and edited by Samuel Kotz. Academic Press, New York, 1970. xx, 196 pp., $12.50. Probability and Mathematical Statistics.

A Mathematical Idea: <i>Lebesgue's Theory of Integration</i> . Its Origins and Development. Thomas Hawkins. University of Wisconsin Press, Madison, 1970. xvi, 228 pp., illus. $12.50.; <i>Classical and Modern Integration Theories</i> . Ivan N. Pesin. Translated from the Russian edition (Moscow, 1966) and edited by Samuel Kotz. Academic Press, New York, 1970. xx, 196 pp., $12.50. Probability and Mathematical Statistics.

The Theory of Lebesgue Measure and Integration.

The Theory of Lebesgue Measure and Integration.

Book Review: The theory of Lebesgue measure and integration

Book Review: The theory of Lebesgue measure and integration