Due February 16
Let with .
- For each , there exists an open interval I such that .
- If E is measurable, the difference set contains an open interval centered at the origin.
Let C be the Cantor set.
- if and only if , where .
- The Cantor-Lebesgue function is defined on C by , if F is well-defined, continuous on C, and , and surjective.
Construct a compact set D in an analogous way as the Cantor set but, at the kth stage of the construction, we remove central open intervals of length , with
- If , then .
- For each , there exists a sequence such that and , where each is one of the removed open intervals in the construction of D, with .
- D is perfect, uncountable, and does not contain any interval.
Let be a sequence of measurable functions on such that a.e. x. Then there exists a sequence such that
Let be separately continuous: continuous in each variable when the other one is fixed. Then f is measurable on .