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Results tagged with measure-theory
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user 10457
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
2
votes
Is there a density theorem for Banach measure?
The answer is no. To see this first observe that if Lebesgue's density theorem holds for $\mu$, then for every set $A$ having $\mu(A)>0$, there is a $\delta>0$ such that for all $t\in \mathbb R^2$ wi …
- 929
5
votes
Accepted
Do sets of big returns contain sets of returns?
No, a counterexample is due to Alan Forrest:
Forrest, A. H., The construction of a set of recurrence which is not a set of strong recurrence, Isr. J. Math. 76, No. 1-2, 215-228 (1991). ZBL0773.28014.
…
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2
votes
Steinhaus theorem and Hausdorff dimension
Lemma 2.7 of [1] says that if $A$ is a compact subset of a separable compact abelian group $G$ with Haar measure 0, then there is a compact set $B\subset G$ with positive Haar measure such that $A+B$ …
- 929
21
votes
Accepted
Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle
Manfred Einsiedler and Alexander Fish have a paper (arxiv.org/abs/0804.3586) showing that a multiplicative subsemigroup of $\mathbb N$ which is not too sparse satisfies the desired measure classificat …
- 929
5
votes
0
answers
130
views
Hartman uniform distribution of means
Background: for a discrete abelian group $G$, a character of $G$ is a homomorphism $\chi:G\to \mathbf S^1$, $\mathbf S^1$ being the circle group $\{z\in \mathbb C:|z|=1\}$ with ordinary multiplication …
- 929