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Results tagged with lebesgue-measure
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user 172802
2
votes
Accepted
Property of sets of positive Lebesgue measure in $\mathbb{R}^2$
Firstly, a set $P$ of positive measure need not contain anything of the form $A\times B$, for example consider for some $k\in\mathbb{R}\setminus\{0\}$ the set $P=\{(x,y)\in\mathbb{R}^2;x-ky\not\in\mat …
- 10.6k
4
votes
0
answers
342
views
If a derivative is defined everywhere and $\pm1$ almost everywhere, is it constant?
Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that the set $A:=\{x\in\mathbb{R};f'(x)\not\in\{1,-1\}\}$ has measure $0$. Does this imply that $f'$ is constant?
Context: I was think …
- 10.6k
11
votes
Accepted
Uncountable collections of distinct subsets of an interval (existence)
My comment reposted as an answer:
If the continuum hypothesis holds, then we can give a well order $\prec$ to $\mathbb{R}$ isomorphic to the first uncountable ordinal. And then for each $j\in[-1,1]$ w …
- 10.6k