Problem 1. Let \( E \) be the closed unit square. Prove that
a) Every open subset of \( E \) is measurable;
b) Every closed subset of \( E \) is measurable;
c) Every set obtained from open and closed subsets of \( E \) by forming no more than a countable number of unions, intersections and complements is measurable.
Comment. There are measurable subsets of \( E \) which are not of the type c).
Problem 2. Construct a theory of Lebesgue measure for sets on the line, starting from intervals (closed, open and half-open) instead of rectangles. Do the same for
a) Sets on the circumference of a circle;
b) Three-dimensional sets;
c) Sets in \( R^{n} \).
Problem 3. Prove that the set of all rational points on the line is measurable, with measure zero.
Problem 4. Prove that the Cantor set constructed in Example 4, p. 52 is measurable, with measure zero.
Problem 5. Prove that every set of positive measure in the interval \( [0,1] \) contains a pair of points whose distance apart is a rational number.
Problem 6. Show that the power of the set of all measurable subsets c the interval \( [0,1] \) is greater than the power of the continuum.
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Problem 7. Let \( C \) be a circle of circumference 1, and let \( \alpha \) be an irrational number. Let all points of \( C \) which can be obtained from each other by rotating \( C \) through an angle \( n \alpha \pi \) (where \( n \) is any integer, positive, negative or zero) be assigned to the same class. (Clearly, each such class contains countably many points.) Let \( \Phi_{0} \) be any set containing one point from each class. Prove that \( \Phi_{0} \) is nonmeasurable.
