Wikipedia:Reference desk/Archives/Mathematics/2006 October 3

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Subsets of R^(n) with empty interior vs those w/ measure zero

Hi, I am so rusty. Is it true that a subset of Euclidean n-space has empty interior if and only if it has (n-space) measure zero? ThanksRich 11:07, 3 October 2006 (UTC)[reply]

Do you mean Lebesgue measure? If so, then no. A cube from which rational points were taken out has an empty interior but positive measure. But the "if" part is true. -- Meni Rosenfeld (talk) 12:17, 3 October 2006 (UTC)[reply]

Cool. Is it possible to say what value that measure takes? -130.237.48.27 12:26, 3 October 2006 (UTC)[reply]

Can you be more specific? In my example, it is the same as that of the original cube. Adding or removing countably many points does not change the Lebesgue measure. -- Meni Rosenfeld (talk) 12:32, 3 October 2006 (UTC)[reply]

(I was 130.237.48.27 for a brief moment.) Ah, you have already answered my question, thank you. —Bromskloss 13:51, 3 October 2006 (UTC)[reply]

Mine as well thanks. Yeah I meant Lebesgue.Rich 16:26, 3 October 2006 (UTC)[reply]