{"title":"纯不可约度量空间与Lipschitz函数的扰动","authors":"David Bate","doi":"10.4310/ACTA.2020.v224.n1.a1","DOIUrl":null,"url":null,"abstract":"We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of the set of all bounded 1-Lipschitz functions $f\\colon X \\to \\mathbb R^m$ with respect to the supremum norm. In one such characterisation it is shown that, if $S$ has positive lower density almost everywhere, then the set of all $f$ with $\\mathcal H^n(f(S))=0$ is residual. Conversely, if $E\\subset X$ is $n$-rectifiable with $\\mathcal H^n(E)>0$, the set of all $f$ with $\\mathcal H^n(f(E))>0$ is residual. \nThese results provide a replacement for the Besicovitch-Federer projection theorem in arbitrary metric spaces, which is known to be false outside of Euclidean spaces.","PeriodicalId":4,"journal":{"name":"ACS Applied Energy Materials","volume":null,"pages":null},"PeriodicalIF":5.4000,"publicationDate":"2017-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"14","resultStr":"{\"title\":\"Purely unrectifiable metric spaces and perturbations of Lipschitz functions\",\"authors\":\"David Bate\",\"doi\":\"10.4310/ACTA.2020.v224.n1.a1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of the set of all bounded 1-Lipschitz functions $f\\\\colon X \\\\to \\\\mathbb R^m$ with respect to the supremum norm. In one such characterisation it is shown that, if $S$ has positive lower density almost everywhere, then the set of all $f$ with $\\\\mathcal H^n(f(S))=0$ is residual. Conversely, if $E\\\\subset X$ is $n$-rectifiable with $\\\\mathcal H^n(E)>0$, the set of all $f$ with $\\\\mathcal H^n(f(E))>0$ is residual. \\nThese results provide a replacement for the Besicovitch-Federer projection theorem in arbitrary metric spaces, which is known to be false outside of Euclidean spaces.\",\"PeriodicalId\":4,\"journal\":{\"name\":\"ACS Applied Energy Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":5.4000,\"publicationDate\":\"2017-12-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"14\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Energy Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4310/ACTA.2020.v224.n1.a1\",\"RegionNum\":3,\"RegionCategory\":\"材料科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"CHEMISTRY, PHYSICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Energy Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/ACTA.2020.v224.n1.a1","RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"CHEMISTRY, PHYSICAL","Score":null,"Total":0}
Purely unrectifiable metric spaces and perturbations of Lipschitz functions
We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of the set of all bounded 1-Lipschitz functions $f\colon X \to \mathbb R^m$ with respect to the supremum norm. In one such characterisation it is shown that, if $S$ has positive lower density almost everywhere, then the set of all $f$ with $\mathcal H^n(f(S))=0$ is residual. Conversely, if $E\subset X$ is $n$-rectifiable with $\mathcal H^n(E)>0$, the set of all $f$ with $\mathcal H^n(f(E))>0$ is residual.
These results provide a replacement for the Besicovitch-Federer projection theorem in arbitrary metric spaces, which is known to be false outside of Euclidean spaces.
期刊介绍:
ACS Applied Energy Materials is an interdisciplinary journal publishing original research covering all aspects of materials, engineering, chemistry, physics and biology relevant to energy conversion and storage. The journal is devoted to reports of new and original experimental and theoretical research of an applied nature that integrate knowledge in the areas of materials, engineering, physics, bioscience, and chemistry into important energy applications.