Measure

2021 ◽  
pp. 51-79
Author(s):  
James Davidson
Keyword(s):  
Measure Space ◽  
Extension Theorem ◽  
Main Topic ◽  
Product Spaces ◽  
Borel Functions ◽  
Nonmeasurable Sets

This chapter examines the concept of a measure space. The main topic is the extension theorem and its ramifications. Nonmeasurable sets are illustrated, and then measure concepts for product spaces introduced. Other topics include measurability under transformations, and Borel functions.

An Elementary Result on Exponential Measure Spaces

1972 ◽  
Vol 15 (2) ◽  
pp. 277-278
Author(s):  
C. Y. Shen
Keyword(s):  
Measure Space ◽  
Measure Zero ◽  
Measure Theory ◽  
Short Note ◽  
Measurable Subset ◽  
Sufficient Condition ◽  
Product Spaces ◽  
Necessary And Sufficient Condition ◽  
Measure Spaces ◽  
Necessary And Sufficient

A simple but useful result in the measure theory for product spaces can be stated as follows:Theorem A. A necessary and sufficient condition that a measurable subset E of X×Y has measure zero is that almost every X-section (or almost every Y-section) has measure zero (see [1, §36]).We will show, in this short note, that a similar result also holds for the exponential of measure spaces. Before proceeding any further, we describe briefly here the exponential construction of a measure space.

scholarly journals Generalized Caratheodory Extension Theorem on Fuzzy Measure Space

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Mehmet Şahin ◽  
Necati Olgun ◽  
F. Talay Akyıldız ◽  
Ali Karakuş
Keyword(s):  
Measure Space ◽  
Extension Theorem ◽  
Fuzzy Measure ◽  
Outer Measure ◽  
Fuzzy Measures ◽  
Set Functions ◽  
Bottom Element ◽  
Entire Universe ◽  
Carathéodory Extension Theorem

Lattice-valued fuzzy measures are lattice-valued set functions which assign the bottom element of the lattice to the empty set and the top element of the lattice to the entire universe, satisfying the additive properties and the property of monotonicity. In this paper, we use the lattice-valued fuzzy measures and outer measure definitions and generalize the Caratheodory extension theorem for lattice-valued fuzzy measures.

Ekottarika-?gama Discourse Without Parallels

2018 ◽  
Vol 35 (1-2) ◽  
pp. 125-134
Author(s):  
Anālayo Bhikkhu
Keyword(s):  
Main Topic ◽  
Pure Land ◽  
Central Importance ◽  
Oral Transmission

With the present paper I study and translate a discourse in the Ekottarika-?gama preserved in Chinese of which no parallel in other discourse collections is known. This situation relates to the wider issue of what significance to accord to the absence of parallels from the viewpoint of the early Buddhist oral transmission. The main topic of the discourse itself is perception of impermanence, which is of central importance in the early Buddhist scheme of the path for cultivating liberating insight. A description of the results of such practice in this Ekottarika-?gama discourse has a somewhat ambivalent formulation that suggests a possible relation to the notion of rebirth in the Pure Abodes, suddh?v?sa. This notion, attested in a P?li discourse, in turn might have provided a precedent for the aspiration, prominent in later Buddhist traditions, to be reborn in the Pure Land.

The 2018/19 beet campaign – Suiker Unie

2019 ◽  
pp. 326-329 ◽  
Author(s):  
Nico Antens
Keyword(s):  
The Netherlands ◽  
Weather Conditions ◽  
Main Topic ◽  
Process Disturbances

As in the rest of Europe beet cultivation and the related beet quality was the main topic at Suiker Unie this campaign due to the very dry weather conditions. With a rather low beet yield as a result it was a short campaign of only 120 days in the Netherlands and 100 days at the Anklam factory in Germany. Despite the unusual beet quality and its related impact on the process, all factories had rather stable beet slicing rates. The beet cultivation and harvest, main investments and process disturbances of the 2018/19 campaign at the three factories Anklam (D), Dinteloord (NL) and Vierverlaten (NL) are discussed.

On the Integrability of the Ergodic Hilbert Transform for A Class of Functions with Equal Absolute Values

1998 ◽  
Vol 5 (2) ◽  
pp. 101-106
Author(s):  
L. Ephremidze
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Integrable Function ◽  
Finite Measure ◽  
Ergodic Transformation ◽  
Finite Measure Space ◽  
Class Of Functions

Abstract It is proved that for an arbitrary non-atomic finite measure space with a measure-preserving ergodic transformation there exists an integrable function f such that the ergodic Hilbert transform of any function equal in absolute values to f is non-integrable.

scholarly journals Fuzzy Inner Product Space: Literature Review and a New Approach

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 765
Author(s):  
Lorena Popa ◽  
Lavinia Sida
Keyword(s):  
Literature Review ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Comprehensive Overview ◽  
New Approach ◽  
Inner Product Spaces ◽  
Product Function ◽  
Suitable Definition

The aim of this paper is to provide a suitable definition for the concept of fuzzy inner product space. In order to achieve this, we firstly focused on various approaches from the already-existent literature. Due to the emergence of various studies on fuzzy inner product spaces, it is necessary to make a comprehensive overview of the published papers on the aforementioned subject in order to facilitate subsequent research. Then we considered another approach to the notion of fuzzy inner product starting from P. Majundar and S.K. Samanta’s definition. In fact, we changed their definition and we proved some new properties of the fuzzy inner product function. We also proved that this fuzzy inner product generates a fuzzy norm of the type Nădăban-Dzitac. Finally, some challenges are given.

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