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I am just curious if there are any measurable(in the sense of Lebesque measure) sets with noncomputable measure just as there are noncomputable real numbers. I would be highly obliged for any clarifications

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    $\begingroup$ Does $[0, \alpha]$ where $\alpha$ is noncomputable work for you? Or are you looking for something else? $\endgroup$
    – Hayden
    May 14 at 5:26
  • $\begingroup$ Meanwhile, while the measure of a "computably closed" set (= $\Pi^0_1$ class) need not be computable, it will always be right c.e., so if you want a closed set of very noncomputable measure you will need to "cheat" a bit. $\endgroup$ May 14 at 5:42
  • $\begingroup$ Thank you for your useful comments $\endgroup$ May 14 at 6:35

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