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Does there exist a characterization of

  • subsets of a segment such that the Lebesgue measure of its $\epsilon$-neighborhood tends to $0$ as $\epsilon\searrow 0$;
  • bounded functions that are continuous on the complement of a set of this type?

I mean either standard terminology or something which could be more convenient to work with.

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    The following paper might also be of interest: Orrin Frink, *Jordan Measure and Riemann Integration*, Annals of Mathematics (2) 34 #3 (July 1933), 518-526. http://math.uga.edu/~pete/Frink33.pdf2011-11-14

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For any bounded set $A$, the limit of Lebesgue measure of its $\epsilon$-neighborhood as $\epsilon \to 0$ gives the Lebesgue measure of the closure $\overline{A}$. This is because the intersection of $\epsilon$-neighborhoods is the closure, and the measure is continuous on decreasing sequences of sets of finite measure. Therefore, the first condition amounts to $m(\overline{A})=0$.

The second condition can be rephrased as: bounded functions that are continuous on an open set of full measure. This property implies Riemann integrabiity, but is not equivalent to it (Thomae's function is Riemann integrable despite having a dense set of discontinuities).