1. Let (S, d) be a complete separable metric space with S non-empty. Sup- pose S has no isolated points, that is, for each x ∈ S, x is in the closure of S\{x }. Prove that there exists a Borel measure µ on S with µ(S) = 1 and µ({x }) = 0 for all x ∈ S. Hint: Define a 1-1 Borel measurable function f from [0, 1] into S and let µ = λ ? f -1.
2. Show that there is a (non-Hausdorff) topology on a set X of two points for which the Baire and Borel σ-algebras are different.
3. A collection L of subsets of a set X will be called a lattice iff /0 ∈ L, X ∈ L, and for any A and B in L, A ∪ B ∈ L and A ∩ B ∈ L. Show that then the collection D of all sets A\B for A and B in L is a semiring (§3.2). Then show that the algebra A generated by (smallest algebra including) a lattice L is the collection of all finite unions l1≤i ≤n Ai \Bi for Ai and Bi in L, where the sets Ai \Bi can be taken to be disjoint for different i. (In any topological space, the collection of all open sets forms a lattice, as does the collection of all closed sets.)