1. For a finite measure space (X, S, µ), suppose there are points xi and numbers ti > 0 such that for any A ∈ S, µ( A) = ),{ti : xi ∈ A}. (Then µ is purely atomic, as defined in §3.5.) Suppose that the singleton {x }∈ S for each x ∈ X . Show that every set in S is regular.
2. Let µ be a finite Borel measure on a separable metric space S. As- sume µ({x }) = 0 for all x ∈ S. Prove that there is a set A of first cat- egory (a countable union of nowhere dense sets; see Theorem 2.5.2) with µ(S\ A) = 0.