In the construction of the Haar measure on a locally compact Hausdorff group $G$ that is standard in the literature, one usually makes a combinatorial definition first: Given a compact set $A$ and an open $U$, one defines $[A : U]$ to be the minimum number of $G$-translates of $U$ needed to cover $A$ (a finite number by compactness of $A$). Then one choses a compact $A_0$ with non-empty interior as scale and defines $$ \lambda_U(A) = \frac{[A : U]}{[A_0 : U]}.$$ Using a suitable choice of ultrafilter one obtains a content $\lambda(A) = \lim_U \lambda_U(A)$ on the set of compacts. Under the usual tricks, this defines a Radon measure $\mu$. One would naively assume that $\mu$ agrees with $\lambda$ on the set of compacts, however, this need not be the case. In general, $\lambda$ need not be regular and it only holds that $$ \mu(A) = \inf_{A \ll A'} \lambda(A')$$ where $A \ll A'$ means there exists an open $U$ such that $A \subset U \subset A'$. This is for example laid out like this in most standard text books, e.g. Fremlin - Measure Theory Section 441, or Halmos - Measure Theory Ch. XI.
I find this inexplicit situation quite unsatisfactory. Could one maybe avoid the choice of an ultrafilter by giving the "right" regular content $\mu(A)$ by some inf-formula directly?
