Mass Problems and Measure-Theoretic Regularity

2009 ◽  
Vol 15 (4) ◽  
pp. 385-409 ◽  
Author(s):  
Stephen G. Simpson
Keyword(s):  
Reverse Mathematics ◽  
Measurable Set ◽  
Borel Hierarchy ◽  
Lebesgue Measurable Set ◽  
Mass Problems

AbstractA well known fact is that every Lebesgue measurable set is regular, i.e., it includes an Fσ set of the same measure. We analyze this fact from a metamathematical or foundational standpoint. We study a family of Muchnik degrees corresponding to measuretheoretic regularity at all levels of the effective Borel hierarchy. We prove some new results concerning Nies's notion of LR-reducibility. We build some ω-models of RCA0 which are relevant for the reverse mathematics of measure-theoretic regularity.

On sets of points of approximate continuity and ϱ-upper continuity

2020 ◽  
Vol 70 (2) ◽  
pp. 305-318
Author(s):  
Anna Kamińska ◽  
Katarzyna Nowakowska ◽  
Małgorzata Turowska
Keyword(s):  
Real Function ◽  
Approximate Continuity ◽  
Measurable Set ◽  
Upper Continuous ◽  
Lebesgue Measurable Set ◽  
Upper Continuity

Abstract In the paper some properties of sets of points of approximate continuity and ϱ-upper continuity are presented. We will show that for every Lebesgue measurable set E ⊂ ℝ there exists a function f : ℝ → ℝ which is approximately (ϱ-upper) continuous exactly at points from E. We also study properties of sets of points at which real function has Denjoy property. Some other related topics are discussed.

Mass Problems and Randomness

2005 ◽  
Vol 11 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Stephen G. Simpson
Keyword(s):  
Computational Complexity ◽  
Positive Measure ◽  
Proof Theory ◽  
Reverse Mathematics ◽  
Equivalence Classes ◽  
Distributive Lattices ◽  
Intermediate Degree ◽  
Associated Mass ◽  
Weakly Reducible ◽  
Mass Problems

AbstractA mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if every member of Q Turing computes a member of P. We say that P is strongly reducible to Q if every member of Q Turing computes a member of P via a fixed Turing functional. The weak degrees and strong degrees are the equivalence classes of mass problems under weak and strong reducibility, respectively. We focus on the countable distributive lattices ω and s of weak and strong degrees of mass problems given by nonempty subsets of 2ω. Using an abstract Gödel/Rosser incompleteness property, we characterize the subsets of 2ω whose associated mass problems are of top degree in ω and s, respectively Let R be the set of Turing oracles which are random in the sense of Martin-Löf, and let r be the weak degree of R. We show that r is a natural intermediate degree within ω. Namely, we characterize r as the unique largest weak degree of a subset of 2ω of positive measure. Within ω we show that r is meet irreducible, does not join to 1, and is incomparable with all weak degrees of nonempty thin perfect subsets of 2ω. In addition, we present other natural examples of intermediate degrees in ω. We relate these examples to reverse mathematics, computational complexity, and Gentzen-style proof theory.

scholarly journals Pointwise density topology

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Magdalena Górajska
Keyword(s):  
Real Line ◽  
Pointwise Convergence ◽  
Baire Property ◽  
Density Topology ◽  
Borel Set ◽  
Convergence In Measure ◽  
The Real ◽  
Measurable Set ◽  
New Type ◽  
Lebesgue Measurable Set

AbstractThe paper presents a new type of density topology on the real line generated by the pointwise convergence, similarly to the classical density topology which is generated by the convergence in measure. Among other things, this paper demonstrates that the set of pointwise density points of a Lebesgue measurable set does not need to be measurable and the set of pointwise density points of a set having the Baire property does not need to have the Baire property. However, the set of pointwise density points of any Borel set is Lebesgue measurable.

Loci of complex polynomials, part II: polar derivatives

2015 ◽  
Vol 159 (2) ◽  
pp. 253-273 ◽  
Author(s):  
BLAGOVEST SENDOV ◽  
HRISTO SENDOV
Keyword(s):  
Higher Order ◽  
Complex Polynomial ◽  
Complex Polynomials ◽  
Point Sets ◽  
Closed Set ◽  
Measurable Set ◽  
Closed Point ◽  
Polar Derivatives ◽  
Lebesgue Measurable Set ◽  
Derivatives Of

AbstractFor every complex polynomial p(z), closed point sets are defined, called loci of p(z). A closed set Ω ⊆ ${\mathbb C}$* is a locus of p(z) if it contains a zero of any of its apolar polynomials and is the smallest such set with respect to inclusion. Using the notion locus, some classical theorems in the geometry of polynomials can be refined. We show that each locus is a Lebesgue measurable set and investigate its intriguing connections with the higher-order polar derivatives of p.

scholarly journals On the p-norm of the truncated Hilbert transform

1988 ◽  
Vol 38 (3) ◽  
pp. 413-420 ◽  
Author(s):  
W. McLean ◽  
D. Elliott
Keyword(s):  
Hilbert Transform ◽  
Positive Measure ◽  
Integral Operators ◽  
Finite Hilbert Transform ◽  
Measurable Set ◽  
Symmetry Properties ◽  
Lebesgue Measurable Set ◽  
The One ◽  
The Hilbert Transform

The p-norm of the Hilbert transform is the same as the p-norm of its truncation to any Lebesgue measurable set with strictly positive measure. This fact follows from two symmetry properties, the joint presence of which is essentially unique to the Hilbert transform. Our result applies, in particular, to the finite Hilbert transform taken over (−1, 1), and to the one-sided Hilbert transform taken over (0, ∞). A related weaker property holds for integral operators with Hardy kernels.

On the lack of equi-measurability for certain sets of Lebesgue-measurable functions

2017 ◽  
Vol 67 (6) ◽  
Author(s):  
Marianna Tavernise ◽  
Alessandro Trombetta ◽  
Giulio Trombetta
Keyword(s):  
Measurable Functions ◽  
Measurable Set ◽  
Lebesgue Measurable Set

AbstractLet Ω be a Lebesgue-measurable set in ℝ

scholarly journals Measurability of functions of two variables

1959 ◽  
Vol 1 (1) ◽  
pp. 21-26 ◽  
Author(s):  
J. H. Michael ◽  
B. C. Rennie
Keyword(s):  
Real Function ◽  
The Other ◽  
Main Result Theorem ◽  
Functions Of Two Variables ◽  
Measurable Set ◽  
Result Theorem ◽  
Lebesgue Measurable Set

SummaryThis paper investigates the existence and equality of the double and repeated integrals of a real function on a plane set. The main result (Theorem 2) is that if a function on a plane Lebesgue measurable set is continuous in one variable and measurable in the other then it is measurable in the plane.

Quantitative continuation from a measurable set of solutions of elliptic equations

2000 ◽  
Vol 130 (4) ◽  
pp. 909-923 ◽  
Author(s):  
S. Vessella
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Positive Measure ◽  
Symmetric Matrix ◽  
Measurable Set ◽  
Connected Set ◽  
Lebesgue Measurable Set

Consider an open bounded connected set Ω in Rn and a Lebesgue measurable set E ⊂⊂ Ω of positive measure. Let u be a solution of the strictly elliptic equation Di (aij Dj u) = 0 in Ω, where aij ∈ C0, 1 (Ω̄) and {aij} is a symmetric matrix. Assume that |u| ≤ ε in E. We quantify the propagation of smallness of u in Ω.

Perturbation of functions by the paths of a Lévy process

1989 ◽  
Vol 105 (2) ◽  
pp. 377-380 ◽  
Author(s):  
Steven N. Evans
Keyword(s):  
Continuous Function ◽  
Lebesgue Measure ◽  
Stable Process ◽  
Levy Process ◽  
Probabilistic Argument ◽  
Scaling Properties ◽  
Measurable Set ◽  
Lebesgue Measurable Set ◽  
Image Position ◽  
Intermediate Value

In a recent paper Mountford [4] showed, using an ingenious probabilistic argument, that if X is a real-valued stable process with index α < 1 and f: [0, ∞) → ℝ is a non-constant continuous function, thenwhere we use the notation |A| for the Lebesgue measure of a Lebesgue measurable set A ⊂ ℝ. The argument in [4] appears to make strong use of the strict scaling properties of X and the ‘intermediate value’ property of f.

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