Measures: Back and Forth Between Point sets and Large sets

1995 ◽  
Vol 1 (2) ◽  
pp. 170-188 ◽  
Author(s):  
Noa Goldring
Keyword(s):  
Lebesgue Measure ◽  
Set Theory ◽  
Real Line ◽  
Measure Theory ◽  
Impossibility Result ◽  
Point Sets ◽  
Large Sets ◽  
Starting Point ◽  
Standard Set ◽  
Action At A Distance

It was questions about points on the real line that initiated the study of set theory. Points paved the way to point sets and these to ever more abstract sets. And there was more: Reflection on structural properties of point sets not only initiated the study of ordinary sets; it also supplied blueprints for defining extra-ordinary, “large” sets, transcending those provided by standard set theory. In return, the existence of such large sets turned out critical to settling open conjectures about point sets.How to explain such action at a distance between the very large and the rather small? Rather than having an air of magic, could these results rest on deep structural similarities between the two superficially distant species of sets?In this essay I dissect one group of such two-way results. Their linchpin is the notion of measure.§1. Vitali's impossibility result. Our starting point is a problem in measure theory regarding the notion of “Lebesgue measure.” Before presenting the problem, I would like to review the notion of Lebesgue measure. Rather than listing its main properties, I would like to show how Lebesgue measure is born out of an attempt to generalize the notion of the length of an interval to arbitrary sets of reals. One tries to approximate arbitrary sets of reals by intervals, in the hope that the lengths of the intervals will induce a measure on these sets.

On ideals of subsets of the plane and on Cohen reals

1986 ◽  
Vol 51 (3) ◽  
pp. 560-569 ◽  
Author(s):  
Jacek Cichoń ◽  
Janusz Pawlikowski
Keyword(s):  
Lebesgue Measure ◽  
Set Theory ◽  
Real Line ◽  
Measure Zero ◽  
Proper Ideal ◽  
Lebesgue Measure Zero ◽  
Borel Set ◽  
Zero Sets ◽  
Models Of Set Theory ◽  
Meager Sets

AbstractLet be any proper ideal of subsets of the real line R which contains all finite subsets of R. We define an ideal * ∣ as follows: X ∈ * ∣ if there exists a Borel set B ⊂ R × R such that X ⊂ B and for any x ∈ R we have {y ∈ R: 〈x, y〉 ∈ B} ∈ . We show that there exists a family ⊂ * ∣ of power ω1 such that ⋃ ∉ * ∣ .In the last section we investigate properties of ideals of Lebesgue measure zero sets and meager sets in Cohen extensions of models of set theory.

scholarly journals Development of an international standard set of patient-centred outcome measures for overall paediatric health: a consensus process

2020 ◽  
pp. archdischild-2020-320345
Author(s):  
Beatrix Algurén ◽  
Jessily P Ramirez ◽  
Matthew Salt ◽  
Nick Sillett ◽  
Stacie N Myers ◽  
...  
Keyword(s):  
Outcome Measures ◽  
Healthcare Delivery ◽  
Well Being ◽  
Delphi Process ◽  
Global Perspective ◽  
Modified Delphi ◽  
Starting Point ◽  
Patient Reported ◽  
Standard Set ◽  
Paediatric Health

ObjectiveTo develop an Overall Pediatric Health Standard Set (OPH-SS) of outcome measures that captures what matters to young people and their families and recognising the biopsychosocial aspects of health for all children and adolescents regardless of health condition.DesignA modified Delphi process.SettingThe International Consortium for Health Outcomes Measurement convened an international Working Group (WG) comprised of 23 international experts from 12 countries in the field of paediatrics, family medicine, psychometrics as well as patient advisors. The WG participated in 11 video-conferences, through a modified Delphi process and 9 surveys between March 2018 and January 2020 consensus was reached on a final recommended health outcome standard set. By a literature review conducted in March 2018, 1136 articles were screened for clinician and patient-reported or proxy-reported outcomes. Further, 4315 clinical trials and 12 paediatric health surveys were scanned. Between November 2019 and January 2020, the final standard set was endorsed by a patient validation (n=270) and a health professional (n=51) survey.ResultsFrom a total of 63 identified outcomes, consensus was formed on a standard set of outcome measures that comprises 10 patient-reported outcomes, 5 clinician-reported measures, and 6 case-mix variables. The four developmental age-specific packages (ie, 0–5, 6–12, 13–17, 18–24 years) include either five or six measures with an average time for completion of 20 min.ConclusionsThe OPH-SS is a starting point to drive value-based paediatric healthcare delivery from a global perspective for enhancing child and adolescent physical health and psychosocial well-being.

scholarly journals Applying Set Theory E88

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 329
Author(s):  
Saharon Shelah
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Real Line ◽  
Cantor Sets ◽  
General Topology ◽  
The Real ◽  
Collectionwise Hausdorff ◽  
Monadic Logic

We prove some results in set theory as applied to general topology and model theory. In particular, we study ℵ1-collectionwise Hausdorff, Chang Conjecture for logics with Malitz-Magidor quantifiers and monadic logic of the real line by odd/even Cantor sets.

Kolmogorov Probability Theory

2017 ◽  
pp. 1-54
Author(s):  
Renáta Bartková ◽  
Beloslav Riečan ◽  
Anna Tirpáková
Keyword(s):  
Probability Theory ◽  
20Th Century ◽  
Set Theory ◽  
Measure Theory ◽  
Basic Ideas

Kolmogorov probability theory based on set theory belongs to the most important results of mathematics of the 20th century. Naturally, its main advantage is the possibility to use results of the modern measure theory. However, this fact sometimes does not allow larger considerations. In this chapter we want to show this paradox can be eliminated. Of course, we present only some basic ideas. Understanding them enables one to study further results and applications.

Category theory, introduction to

2018 ◽  
Author(s):  
Colin McLarty
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Category Theory ◽  
Standard Set ◽  
Universe Of Sets ◽  
The Universe

A ‘category’, in the mathematical sense, is a universe of structures and transformations. Category theory treats such a universe simply in terms of the network of transformations. For example, categorical set theory deals with the universe of sets and functions without saying what is in any set, or what any function ‘does to’ anything in its domain; it only talks about the patterns of functions that occur between sets. This stress on patterns of functions originally served to clarify certain working techniques in topology. Grothendieck extended those techniques to number theory, in part by defining a kind of category which could itself represent a space. He called such a category a ‘topos’. It turned out that a topos could also be seen as a category rich enough to do all the usual constructions of set-theoretic mathematics, but that may get very different results from standard set theory.

Decidability of the “almost all” theory of degrees

1972 ◽  
Vol 37 (3) ◽  
pp. 501-506 ◽  
Author(s):  
John Stillwell
Keyword(s):  
Lebesgue Measure ◽  
Measure Theory ◽  
Random Sequence ◽  
Recursion Theory ◽  
Sharp Contrast ◽  
Standard Theory ◽  
Simple Manner ◽  
Fubini’S Theorem ◽  
Almost All ◽  
Random Pair

Ever since Spector's brilliant application of measure theory to recursion theory in 1958 [6] it has been realized that measure theory promotes sweeping simplifications in the theory of degrees. Results previously thought to be pathological were shown by Spector, and later Sacks [4], [5], to hold for almost all degrees (“almost all” in the sense of Lebesgue measure), often with much simpler proofs. Good examples of this phenomenon are Spector's demonstration that almost all pairs of sets are of incomparable degree (as an immediate consequence of Fubini's theorem) and Sacks' exquisitely simple deduction from this result that almost every degree is the join of two incomparable degrees (for if a random sequence is decomposed into its even and odd parts, the result is a random pair).The present paper attempts to vindicate the feeling that almost all degrees behave in a simple manner by showing that if the quantifier in the theory of degrees with ′(jump), ∪ (join) and ∩ (meet) is taken to be (almost ∀a) instead of (∀a) then the theory is decidable. We are able to use ∩ because it will be shown that if t1, t2 are any terms built from degree variables a1, …, am with ′ and ∪ then t1 ∩ t2 exists for almost all a1, …, am. Thus the “almost all” theory presents a sharp contrast to the standard theory, where ∩ is not always defined (Kleene-Post [1]) and which is known to be undecidable (Lachlan [2]).

Multifractal formalism for the inverse of random weak Gibbs measures

2019 ◽  
Vol 20 (04) ◽  
pp. 2050024
Author(s):  
Zhihui Yuan
Keyword(s):  
Probability Measure ◽  
Lebesgue Measure ◽  
Real Line ◽  
Gibbs Measures ◽  
Borel Probability Measure ◽  
Cantor Set ◽  
Multifractal Formalism ◽  
The Real ◽  
Random Dynamics ◽  
Borel Probability

Any Borel probability measure supported on a Cantor set included in [Formula: see text] and of zero Lebesgue measure on the real line possesses a discrete inverse measure. We study the validity of the multifractal formalism for the inverse measures of random weak Gibbs measures. The study requires, in particular, to develop in this context of random dynamics a suitable version of the results known for heterogeneous ubiquity associated with deterministic Gibbs measures.

Sojourns and extremes of a diffusion process on a fixed interval

1982 ◽  
Vol 14 (4) ◽  
pp. 811-832 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Diffusion Process ◽  
Lebesgue Measure ◽  
Stationary Distribution ◽  
Real Line ◽  
Diffusion Coefficients ◽  
Limit Theorems ◽  
Random Variable ◽  
Fixed Interval ◽  
The Real ◽  
And Diffusion

Let X(t), , be an Ito diffusion process on the real line. For u > 0 and t > 0, let Lt(u) be the Lebesgue measure of the set . Limit theorems are obtained for (i) the distribution of Lt(u) for u → ∞and fixed t, and (ii) the tail of the distribution of the random variable max[0, t]X(s). The conditions on the process are stated in terms of the drift and diffusion coefficients. These conditions imply the existence of a stationary distribution for the process.

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