Homework 1

Due October 22

1. Complete the proof of Vitali's covering theorem for Radon measures and unbounded sets A.

2. Show an example (a set, a measure, and a family of balls) where the set A in Vitali's covering theorem can only be covered up to measure 0, and cannot be covered in a set theoretic sense.

3. Show that the upper and lower densities of \mu with resepect to \lambda are measurable for the case where \lambda is either Lebesgue measure or the s-dimensional Hausdorff measure.

4. ex. 4 on page 43 of Mattila.

5. ex. 6 on page 53 of Mattila.