Math 6210 • Real Analysis

Let \(X\) be a compact Hausdorff space.

(a) Prove that a complex measure \(\mu\) on \(X\) is positive if and only if
\(\mu(X)=|\mu|(X)\).

(b) Let \(A\subset C(X)\) be a linear subspace containing the function \(1\). Let
\(S\subset X\) be a compact subset with the property that for each \(f\in A\),
\[
\sup\{|f(x)|:x\in X\}=\sup\{|f(x)|:x\in S\}
\]
Prove that for each \(x\in X\) there exists a positive measure \(\mu_x\) with
support contained in \(S\) such that
\[
f(x)=\int_S f\,d\mu_x\qquad\text{for }f\in A.
\]

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