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An Introduction to Measure Theory

Exercise 1.6.26

(i): Give an example of a compact set $K \subset R$ of positive measure such that $m (K \cap I) < | I |$ for every interval $I$ of positive length.
(ii): Give an example of a measurable set $E \subset R$ such that $0 < m (E \cap I) < | I |$ for every interval $I$ of positive length.