Due February 9
The Cantor set is totally disconnected (for any there is between x and y) and perfect (compact and without isolated points).
- If E is compact,
- The previous conclusion may be false for E closed and unbounded, or open and bounded.
Problem 3 (Borel-Cantelli lemma)
Let be a sequence of measurable sets such that
and define for infinitely many .
- E is measurable.
Let, for a subset ,
where the infimum is taken over all countable covers for E of rectangles. Then
We can thus conclude that we obtain the same Lebesgue measure theory if cubes are replaced by rectangles.