Math 6210 • Real Analysis

Let \(\mathcal F=\{f_\alpha \}_{\alpha \in A}\) be a family of continuous complex valued functions on a compact Hausdorff space. Let \(\{c_\alpha \}_{\alpha \in A}\) be a family of complex numbers. Suppose that for each \(F\subset A\) with \(F\) finite, there exists a complex measure \(\mu _F\) of total variation one on \(X\) such that \[ \int _X f_\alpha \,d\mu _F=c_\alpha \qquad (\forall \alpha \in F). \] Show that there exists a complex measure \(\mu \) on \(X\) of total variation at most one such that \[ \int _X f_\alpha \,d\mu =c_\alpha \qquad (\forall \alpha \in A). \]