Exercise 4, § 2

Let A be an uncountable set. In the set $\mathfrak{P}(A)$, consider the relation « M$\cap \mathbf{C} N$ and $N \cap \mathbf{C} M$ are countable » between M and N; show that this is an equivalence relation, weakly compatible with the inclusion relation $M \subset M'$ (S, III, §1, Exer. 2). Let B be the quotient set of $\mathfrak{P}(A)$ for this equivalence relation; show that on B, the relation deduced from the inclusion relation by passage to quotients is an order relation for which B is a Boolean algebra (loc. cit, Exer. 17). One knows (GT, II, §4, Exer. 12) that there exists an order structure isomorphism of B onto the Boolean algebra formed by the subsets that are both open and closed in a totally disconnected compact space X. Show that if U is a nonempty open set in X, then there exists an uncountable family of pairwise disjoint nonempty open subsets $V_\alpha$ of U (make use of the fact that an uncountable set admits an uncountable partition consisting of uncountable sets). From this, deduce that every measure on X has nowhere dense support.