Search results for "Borel set"
showing 6 items of 6 documents
A space on which diameter-type packing measure is not Borel regular
1999
We construct a separable metric space on which 1-dimensional diameter-type packing measure is not Borel regular.
A weak comparison principle for solutions of very degenerate elliptic equations
2012
We prove a comparison principle for weak solutions of elliptic quasilinear equations in divergence form whose ellipticity constants degenerate at every point where \(\nabla u\in K\), where \(K\subset \mathbb{R }^N\) is a Borel set containing the origin.
On the conical density properties of measures on $\mathbb{R}^n$
2005
We compare conical density properties and spherical density properties for general Borel measures on $\mathbb{R}^n$ . As a consequence, we obtain results for packing and Hausdorff measures $\mathcal{P}_h$ and $\mathcal{H}_h$ provided that the gauge function $h$ satisfies certain conditions. One consequence of our general results is the following: let $m, n\,{\in}\,\mathbb{N}, 0\,{\lt}\,s\,{\lt}\,m\,{\leq}\,n$ , $0\,{\lt}\,\eta\,{\lt}\,1$ , and suppose that $V$ is an $m$ -dimensional linear subspace of $\mathbb{R}^n$ . Let $\mu$ be either the $s$ -dimensional Hausdorff measure or the $s$ -dimensional packing measure restricted to a set $A$ with $\mu(A)\,{\lt}\,\infty$ . Then for $\mu$ -almos…
A Note on Algebraic Sums of Subsets of the Real Line
2002
AbstractWe investigate the algebraic sums of sets for a large class of invari-ant ˙-ideals and ˙- elds of subsets of the real line. We give a simpleexample of two Borel subsets of the real line such that its algebraicsum is not a Borel set. Next we show a similar result to Proposition 2from A. Kharazishvili paper [4]. Our results are obtained for ideals withcoanalytical bases. 1 Introduction We shall work in ZFC set theory. By !we denote natural numbers. By 4wedenote the symmetric di erence of sets. The cardinality of a set Xwe denoteby jXj. By R we denote the real line and by Q we denote rational numbers. IfAand Bare subsets of R n and b2R , then A+B= fa+b: a2A^b2Bgand A+ b= A+ fbg. Simila…
Singular quasisymmetric mappings in dimensions two and greater
2018
For all $n \geq 2$, we construct a metric space $(X,d)$ and a quasisymmetric mapping $f\colon [0,1]^n \rightarrow X$ with the property that $f^{-1}$ is not absolutely continuous with respect to the Hausdorff $n$-measure on $X$. That is, there exists a Borel set $E \subset [0,1]^n$ with Lebesgue measure $|E|>0$ such that $f(E)$ has Hausdorff $n$-measure zero. The construction may be carried out so that $X$ has finite Hausdorff $n$-measure and $|E|$ is arbitrarily close to 1, or so that $|E| = 1$. This gives a negative answer to a question of Heinonen and Semmes.
On the inverse absolute continuity of quasiconformal mappings on hypersurfaces
2018
We construct quasiconformal mappings $f\colon \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ for which there is a Borel set $E \subset \mathbb{R}^2 \times \{0\}$ of positive Lebesgue $2$-measure whose image $f(E)$ has Hausdorff $2$-measure zero. This gives a solution to the open problem of inverse absolute continuity of quasiconformal mappings on hypersurfaces, attributed to Gehring. By implication, our result also answers questions of V\"ais\"al\"a and Astala--Bonk--Heinonen.