Is it true that in the Riesz representation theorem
$\mu(F)=\sup\{\Lambda(f): f\in C_c, 0\leq f \leq 1, \operatorname{supp} f \subset F \}$
for every compact (or closed) subset $F$?
(It is known that it holds if $F$ is open, and that $\mu(F)=\inf\{\Lambda(f): f\in C_c, 0\leq f \leq 1, f(x)=1 \ for \ x \in F \}$ if $F$ is compact.) I think it may be true, because in "Abstract harmonic analysis", vol.I., by E. Hewitt and K. Ross (if I well understood) the measure in the representation theorem is defined for closed subsets just in such a way (chap.11, (def. 11.20, th. 11.17 (i), def.11.11).
Thanks