Can you give examples of the following sets, please?

Question

1) A subset of R which is meager in Baire classification and has positive measure;

2) A subset of R with has null measure but is not meager.

Thank you

Answer 1

(1) This example made me remember a question I asked some days ago. Let r_n be an enumeration of the rationals and let a_n be a sequence of positive numbers such that Sum a_n converges. Let A = Union [(r_n - a_n , r_n + b_n)]. Then, A is open (union of open intervals), is dense in R (contains the rationals) and is not all of R, because it has finite measure.

Now, let B be the complement of A. Then, A is closed, has an empty interior (because A is dense) and has infinite measure. Hence, B is not only meager, but nowhere dense, and has infinite measure!

If you want finite measure, do something similar enumerating the rationals in [0, 1].

(2) I don't have an specific example right now, but I'm sure there are "fat" Cantor sets. They are not meager and are null. I'll try to find an specific example.