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Results tagged with measure-theory
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user 5371
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
12
votes
Projection of Borel set from $R^2$ to $R^1$
This was a famous mistake made by Lebesgue (see also Gerald Edgar's answer to this MO question).
Suslin showed that a plane Borel set exists whose projection is not a Borel set.
See the references to …
- 22.3k
6
votes
Is Conway's base-13 function measurable?
It seems that the definition of the function doesn't use the axiom of choice. This implies that the support set should be Lebesgue measurable.
- 22.3k
4
votes
Regular borel measures on metric spaces
Let X be a metric space. Then every Borel measure μ on X is regular (i.e. for every Borel set B and every ε > 0, there exists a closed set $F_ε$ such that $F_ε ⊂ B$ and μ(B\ $F_ε$) < ε). If X is comp …
- 22.3k
5
votes
A question about the Kakeya problem
A Kakeya needle set cannot be of measure zero (a line segment cannot be rotated continuously within a set of measure 0). See the blog post by Terry Tao.
However, there are sets of measure zero withi …
- 22.3k
20
votes
Accepted
Borel set plus a closed set = Borel
No. Erdös and Stone showed that the sum of two subsets $E$, $F\subset\mathbb R$ may not be Borel even if one of them is compact and the other is $G_\delta$ (see "On the Sum of Two Borel Sets", Proc. A …
- 22.3k
11
votes
A set that can be covered by arbitrarily small intervals
The problem was also studied by Besicovitch from the geometric measure-theoretic point of view in the 1930s. In particular, Besicovitch was motivated by the problem of determining the sets of reals on …
- 22.3k