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Relevant bibliographies by topics / Lebesgue outer measure

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Academic literature on the topic 'Lebesgue outer measure'

Author: Grafiati

Published: 4 June 2021

Last updated: 14 February 2022

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Journal articles on the topic "Lebesgue outer measure"

1

Endou, Noboru. "Reconstruction of the One-Dimensional Lebesgue Measure." Formalized Mathematics 28, no. 1 (2020): 93–104. http://dx.doi.org/10.2478/forma-2020-0008.

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SummaryIn the Mizar system ([1], [2]), Józef Białas has already given the one-dimensional Lebesgue measure [4]. However, the measure introduced by Białas limited the outer measure to a field with finite additivity. So, although it satisfies the nature of the measure, it cannot specify the length of measurable sets and also it cannot determine what kind of set is a measurable set. From the above, the authors first determined the length of the interval by the outer measure. Specifically, we used the compactness of the real space. Next, we constructed the pre-measure by limiting the outer measure

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2

Taylor, S. James, and Claude Tricot. "The packing measure of rectifiable subsets of the plane." Mathematical Proceedings of the Cambridge Philosophical Society 99, no. 2 (1986): 285–96. http://dx.doi.org/10.1017/s0305004100064203.

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It is usual to define Lebesgue outer measure in ℝ by using economical coverings by a sequence of open intervals. We start by outlining a different definition which gives the same answer for a bounded measurable E ⊂ ℝ. PutThen λ0 defines a pre-measure, but is not an outer measure because it is not countably sub-additive. However it leads to an outer measure by definingand it can be proved directly that λ is just another definition of Lebesgue outer measure. We do not give the details of this proof as it can also be deduced as a corollary of the main results in the present paper.

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3

Sarkhel, D. N., and T. Chakraborti. "Topological vitali measure spaces." Bulletin of the Australian Mathematical Society 32, no. 2 (1985): 225–49. http://dx.doi.org/10.1017/s0004972700009928.

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The properties of Lebesgue outer measures embodied in the Vitali covering theorem, the Vitali-Carathéodory theorem, the Lusin theorem, the density theorem, outer regularity and inner regularity, and the relation between measurability and approximate continuity are studied in a general abstract space, called a topological Vitali measure space. The main theme is the mutual equivalence of these properties.

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4

Saputro, Bayu Riyadhus, and Sumaji . "LEBESGUE’S CRITERION FOR INTEGRABILITY." EDUPEDIA 2, no. 1 (2018): 12. http://dx.doi.org/10.24269/ed.v2i1.85.

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Lebesgue integral was defined constructively by using outer measure, measurable set, measurable function and Lebesgue measure concept. Then it was constructed by using simple function. The requirement of a Lebesgue integrable function to be Riemann integrable is bounded function and the set of discontinuities of that function has measure zero.

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5

Leader. "Variation of f on E and Lebesgue Outer Measure of f E." Real Analysis Exchange 16, no. 2 (1990): 508. http://dx.doi.org/10.2307/44153729.

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6

Hejduk, J., and A. Loranty. "On a strong generalized topology with respect to the outer Lebesgue measure." Acta Mathematica Hungarica 163, no. 1 (2021): 18–28. http://dx.doi.org/10.1007/s10474-020-01124-4.

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7

Singh, Jitender. "A Short Proof That Lebesgue Outer Measure of an Interval Is Its Length." American Mathematical Monthly 125, no. 6 (2018): 553. http://dx.doi.org/10.1080/00029890.2018.1452512.

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8

Wojas, Włodzimierz, Jan Krupa, and Jarosław Bojarski. "Familiarizing Students with Definition of Lebesgue Outer Measure Using Mathematica: Some Examples of Calculation Directly from Its Definition." Mathematics in Computer Science 14, no. 2 (2019): 253–70. http://dx.doi.org/10.1007/s11786-019-00435-2.

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AbstractIn this paper we present some examples of calculation the Lebesgue outer measure of some subsets of $$\mathbb {R}^2$$R2 directly from definition 1. We will consider the following subsets of $$\mathbb {R}^2$$R2: $$\displaystyle \{(x, y)\in \mathbb {R}^2: 0\le y \le x^2, x\in [0, 1]\}$${(x,y)∈R2:0≤y≤x2,x∈[0,1]}, $$\displaystyle \{(x, y)\in \mathbb {R}^2: 0\le y \le \exp (-x), x\ge 0\}$${(x,y)∈R2:0≤y≤exp(-x),x≥0}, $$\big \{ (x,y) \in \mathbb {R}^2: \ln x \le y \le 0, x\in (0, 1]\big \}$${(x,y)∈R2:lnx≤y≤0,x∈(0,1]}, $$\big \{ (x,y) \in \mathbb {R}^2: 0\le y \le 1/x, x\ge 1\big \}$${(x,y)∈R2

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9

Judah, Haim, and Saharon Shelah. "The Kunen-Miller chart (Lebesgue measure, the Baire property, Laver reals and preservation theorems for forcing)." Journal of Symbolic Logic 55, no. 3 (1990): 909–27. http://dx.doi.org/10.2307/2274464.

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AbstractIn this work we give a complete answer as to the possible implications between some natural properties of Lebesgue measure and the Baire property. For this we prove general preservation theorems for forcing notions. Thus we answer a decade-old problem of J. Baumgartner and answer the last three open questions of the Kunen-Miller chart about measure and category. Explicitly, in §1: (i) We prove that if we add a Laver real, then the old reals have outer measure one. (ii) We prove a preservation theorem for countable-support forcing notions, and using this theorem we prove (iii) If we add

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Shelah, S., and D. H. Fremlin. "Pointwise compact and stable sets of measurable functions." Journal of Symbolic Logic 58, no. 2 (1993): 435–55. http://dx.doi.org/10.2307/2275214.

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In a series of papers culminating in [9], M. Talagrand, the second author, and others investigated at length the properties and structure of pointwise compact sets of measurable functions. A number of problems, interesting in themselves and important for the theory of Pettis integration, were solved subject to various special axioms. It was left unclear just how far the special axioms were necessary. In particular, several results depended on the fact that it is consistent to suppose that every countable relatively pointwise compact set of Lebesgue measurable functions is ‘stable’ in Talagrand

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Dissertations / Theses on the topic "Lebesgue outer measure"

1

David, Manolis. "The Henstock–Kurzweil Integral." Thesis, Linköpings universitet, Matematik och tillämpad matematik, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-166430.

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Since the introduction of the Riemann integral in the middle of the nineteenth century, integration theory has been subject to significant breakthroughs on a relatively frequent basis. We have now reached a point where integration theory has been thoroughly researched to a point where one has to delve quite deep into a particular subject in order to encounter open conjectures. In education the Riemann integral has for quite some time been the standard integral in elementary analysis courses and as the complexity of these courses incrementally increase the more general Lebesgue integral eventua

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2

Brooks, Hannalie Helena. "Measurable functions and Lebesgue integration." Diss., 2002. http://hdl.handle.net/10500/950.

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In this thesis we shall examine the role of measurability in the theory of Lebesgue Integration. This shall be done in the context of the real line where we define the notion of an integral of a bouuded real-valued function over a set of bounded outer measure without a prior assumption of measurability concerning the function and the domain of integration

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Book chapters on the topic "Lebesgue outer measure"

1

Krantz, Steven G. "Additivity for Outer Measure." In Elementary Introduction to the Lebesgue Integral. Chapman and Hall/CRC, 2018. http://dx.doi.org/10.1201/9781351056823-14.

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Krantz, Steven G. "The Concept of Outer Measure." In Elementary Introduction to the Lebesgue Integral. Chapman and Hall/CRC, 2018. http://dx.doi.org/10.1201/9781351056823-6.

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Chandrasekharan, K. "The outer measure and its applications: the Lebesgue measure." In Texts and Readings in Mathematics. Hindustan Book Agency, 1996. http://dx.doi.org/10.1007/978-93-80250-88-5_3.

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4

"Lebesgue Measure and Outer Measure." In Measure and Integral. Chapman and Hall/CRC, 2015. http://dx.doi.org/10.1201/b18361-9.

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5

"The Outer Measure." In The Elements of Integration and Lebesgue Measure. John Wiley & Sons, Inc., 2011. http://dx.doi.org/10.1002/9781118164471.ch12.

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