 Full support of a faithful spectral measure

For a faithful representation of $C(K)$ by integration against a regular projection-valued measure $P$, every nonempty open subset $U$ satisfies $P(U)\ne 0$. Choose a nonzero continuous function supported inside $U$, using compact Hausdorff normality. If $P(U)=0$, its spectral integral vanishes, contradicting faithfulness. Regularity is understood through the scalar spectral measures.