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日期：2025-11-11 05:14

4. (a) The standard middle quarters Cantor set is defined by removing the middle quarter of from [0,1] then removing the middle quarters of the two remaining pieces and then iterating. Give a fully precise account of this construction. Show this set is a zero set.

(b) We now define a fat Cantor set by modifying the above. First remove the middle interval of length 1/4 from [0, 1] and then remove the middle intervals of length from each of the remaining pieces and then iterate. (At the next step the relevant number is 1/64) Give a fully precise account of this construction. Show that the resulting set is not a zero set.

5. Prove that the integral of the Zeno's staircase function is in fact 2/3.