The problem is from the proof of Theorem 1.5.12 in Leon Simon's book: Geometric Measure Theory
Suppose $X$ is a locally compact Hausdorff space, $\mathcal{K}^{+}$ is the set of all non-negative continuous functions on $X$ with compact support, $\lambda:\mathcal{K}^{+}\to [0,\infty)$ is linear. $U$ open in $X$, $K$ compact in $X$, $A\subset X$. we define$$\mu(U)=\sup\{\lambda(f):f\in\mathcal{K}^{+},f\leq 1,supp(f)\subset U\}$$$$\mu(A)=\inf\{\mu(U):U\mbox{ open},U\supset A\}$$Prove that $\mu(U) = \sup\{\mu(K):K\mbox{ compact}$, $K\subset U\}$.
In Leon Simon's book, the proof of this statement is too short to understand, so I was looking for clearer and more rigorous proof. Obviously LHS $\geq$ RHS, however I cannot prove the equality. Any help will be appreciated.