Tangent measures of typical measures

Tuomas Sahlsten

Tangent measures of typical measures

Tuomas Sahlsten

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that for a typical Radon measure μ in ℝd, every non-zero Radon measure is a tangent measure of μ at μ almost every point. This was already shown by T. O'Neil in his Ph.D. thesis from 1994, but we provide a different self-contained proof for this fact. Moreover, we show that this result is sharp: for any non-zero measure we construct a point in its support where the set of tangent measures does not contain all non-zero measures. We also study a concept similar to tangent measures on trees, micromeasures, and show an analogous typical property for them. Mathematical Reviews subject classification: Primary: 28A12; Secondary: 28A75, 28A80 Key words: tangent measures, Baire category

Original language	English
Pages (from-to)	53-80
Number of pages	28
Journal	Real Analysis Exchange
Volume	40
Issue number	1
Publication status	Published - 2015

Keywords

math.CA
28A12, 28A75, 28A80
Cubes
Tangents
Mathematical theorems

Cite this

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journal = "Real Analysis Exchange",

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AU - Sahlsten, Tuomas

PY - 2015

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N2 - We prove that for a typical Radon measure μ in ℝd, every non-zero Radon measure is a tangent measure of μ at μ almost every point. This was already shown by T. O'Neil in his Ph.D. thesis from 1994, but we provide a different self-contained proof for this fact. Moreover, we show that this result is sharp: for any non-zero measure we construct a point in its support where the set of tangent measures does not contain all non-zero measures. We also study a concept similar to tangent measures on trees, micromeasures, and show an analogous typical property for them. Mathematical Reviews subject classification: Primary: 28A12; Secondary: 28A75, 28A80 Key words: tangent measures, Baire category

AB - We prove that for a typical Radon measure μ in ℝd, every non-zero Radon measure is a tangent measure of μ at μ almost every point. This was already shown by T. O'Neil in his Ph.D. thesis from 1994, but we provide a different self-contained proof for this fact. Moreover, we show that this result is sharp: for any non-zero measure we construct a point in its support where the set of tangent measures does not contain all non-zero measures. We also study a concept similar to tangent measures on trees, micromeasures, and show an analogous typical property for them. Mathematical Reviews subject classification: Primary: 28A12; Secondary: 28A75, 28A80 Key words: tangent measures, Baire category

KW - math.CA

KW - 28A12, 28A75, 28A80

KW - Cubes

KW - Tangents

KW - Mathematical theorems

M3 - Article

VL - 40

SP - 53

EP - 80

JO - Real Analysis Exchange

JF - Real Analysis Exchange

IS - 1

ER -

Tangent measures of typical measures

Abstract

Keywords

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