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Results tagged with lebesgue-measure
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user 165275
3
votes
Why is Lebesgue measure theory asymmetric?
The asymmetry has only historical reasons. It is possible to develop Lebesgue theory (moreover, all extension theorems and thus the theory of product measures) from the “inner approach”. This was done …
- 1,869
7
votes
Accepted
To show a set is a set of positive Lebesgue measure in $ \mathbb{R}$
No. Every set $E$ without interior points (e.g. the complements of the rationals) has the property that $$\bigcap_{|t|<\varepsilon}(t+E)=\emptyset$$ for every $\varepsilon>0$. Indeed, for every $x\in …
- 1,869
7
votes
Accepted
Regarding a positive Lebesgue measure set in $\mathbb{R}^2$
The question was answered by Robert Israel 1995 on Usenet, essentially by the set mentioned in fedja's comment. The proof that this set has the required property is carried out in detail in Example 4. …
- 1,869