Impossibility of decoding a translation invariant measure from a single set of positive Lebesgue measure
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910763335024640 |
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| author | Bulj, Aleksandar |
| author_facet | Bulj, Aleksandar |
| contents | Let $μ$ be a translation invariant measure on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$ and let $λ$ denote the Lebesgue measure on $\mathbb{R}^d$. If there exists an open set $U$ such that $0<μ(U)=λ(U)<\infty$, it is a simple exercise to show that $μ=λ|_{\mathcal{B}(\mathbb{R}^d)}$. Is the same conclusion true if $U$ is merely a Borel set?
The main purpose of this short note is to construct a measure that provides a negative answer to this question. Incidentally, this construction provides a new example of a translation invariant measure with a rich domain and range that is not Hausdorff, a problem previously studied by Hirst. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06065 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Impossibility of decoding a translation invariant measure from a single set of positive Lebesgue measure Bulj, Aleksandar Classical Analysis and ODEs Let $μ$ be a translation invariant measure on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$ and let $λ$ denote the Lebesgue measure on $\mathbb{R}^d$. If there exists an open set $U$ such that $0<μ(U)=λ(U)<\infty$, it is a simple exercise to show that $μ=λ|_{\mathcal{B}(\mathbb{R}^d)}$. Is the same conclusion true if $U$ is merely a Borel set? The main purpose of this short note is to construct a measure that provides a negative answer to this question. Incidentally, this construction provides a new example of a translation invariant measure with a rich domain and range that is not Hausdorff, a problem previously studied by Hirst. |
| title | Impossibility of decoding a translation invariant measure from a single set of positive Lebesgue measure |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2406.06065 |