Problem 2
Let \(\mu\) be a Radon measure on \(X\). a. Let \(N\) be the union of all open \(U \subset X\) such that \(\mu(U)=0\). Then \(N\) is open and \(\mu(N)=0\). The complement of \(N\) is called the support of \(\mu\). b. \(x \in \operatorname{supp}(\mu)\) iff \(\int f d \mu>0\) for every \(f \in C_{c}(X,[0,1])\) such that \(f(x)>0\).
Problem 6
Let \(X\) be an uncountable set with the discrete topology, or the one-point compactification of such a set. Then \(\mathcal{B}_{X} \neq \mathbb{B}_{X}\).
Problem 11
Suppose that \(\mu\) is a Radon measure on \(X\) such that \(\mu(\\{x\\})=0\) for all \(x \in X\), and \(A \in B_{X}\) satisfies \(0<\mu(A)<\infty\). Then for any \(\alpha\) such that \(0<\alpha<\mu(A)\) there is a Borel set \(B \subset A\) such that \(\mu(B)=\alpha\).
Problem 17
If \(\mu\) is a positive Radon measure on \(X\) with \(\mu(X)=\infty\), there exists \(f \in C_{0}(X)\) such that \(\int f d \mu=\infty\). Consequently, every positive linear functional on \(C_{0}(X)\) is bounded.
