Intereting Posts

Traces of all positive powers of a matrix are zero implies it is nilpotent
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How to integrate $\sec^3 x \, dx$?
Question about the proof of $S^3/\mathbb{Z}_2 \cong SO(3)$
An element of $f$ of a function field such that $P$ is the only pole of $f$.
Why do certain notations differ around the world?
Why is a linear transformation a $(1,1)$ tensor?
How do you prove this integral involving the Glaisherâ€“Kinkelin constant
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Is there an accepted term for those objects of a category $X$ such that for all $Y$, there is at most one arrow $X \rightarrow Y$?
Describe all extension groups of a given subgroup $H \trianglelefteq$ Aff$\mathbb{(F_q)}$ by Aff$\mathbb{(F_q)}/H$
Is order of variables important in probability chain rule
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Chebyshev: Proof $\prod \limits_{p \leq 2k}{\;} p > 2^k$
The Power of Taylor Series

Does there exist $K \subseteq \mathbb{R} \backslash \mathbb{Q}$ such that $K$ is compact, and has Lebesgue measure greater than $0$? As I have been trying to think of examples, I suspect that any subset of $\mathbb{R} \backslash \mathbb{Q}$ that is closed can be at most countable, since the closure of an uncountable subset of irrationals should contain some rationals. And, the Lebesgue measure of a countable set is $0$. If there are any examples of such a set, I would be very interested to know how it is constructed.

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The answer is yes. Count the rationals in $[0,1]$ as $r_1,r_2,\ldots$, let $I_k$ be an open interval containing $r_k$ of length $3^{-k}$, and let $K=[0,1]\setminus\cup_k I_k$.

This question is somewhat related to the question Perfect set without rationals, but there measure did not come up. For example, the Cantor set-like construction given there by JDH could be made to have positive measure.

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