# Measurable sets with non computable measure

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

• Does $[0, \alpha]$ where $\alpha$ is noncomputable work for you? Or are you looking for something else? May 14 at 5:26
• 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. May 14 at 5:42
• Thank you for your useful comments May 14 at 6:35