Riesz representation on compactly supported continuous functions ($I(f)={\int }_{X}fd\mu$)

A positive linear functional on the space of continuous compactly supported functions of a locally compact Hausdorff space has a unique representing positive Radon measure. Positivity gives boundedness on fixed compact supports: choose a compactly supported continuous cutoff $p\ge 1$ on the support $K$, then $|I(f)|\le \Vert f{\Vert }_{\infty }I(p)$ for real $f$ supported in $K$. The resulting measure is finite on compact sets but need not have finite total mass.