Weil's construction of a Haar measure on a locally compact group rests on approximating a function from above by sums of translates of another function. I would need to know something similar for an approximation from below. Actually, I need something stronger:

Let $G$ be a locally compact group and let $C_c^+(G)$ be the set of all compactly supported functions $f$ on $G$ with $f\ne 0$ and $f(x)\ge 0$ for all $x\in G$. For $\phi,f\in C_c^+(G)$ let $(\phi,f)$ be the infimum over all sums $\sum_{j=1}^nc_j$, where $c_j>0$ such that there exist $s_1,\dots, s_n\in G$ with $$ \phi(x)\le\sum_{j=1}^nc_j\ f(s_jx),\quad \forall\ x\in G. $$ Likewise, let $[\phi,f]$ be the supremum over all $\sum_{j=1}^nc_j$ such that there exist $s_1,\dots, s_n\in G$ with $$ \phi(x)\ge\sum_{j=1}^nc_j\ f(s_jx),\quad \forall\ x\in G. $$ Then Weil's proof of the existence and uniqueness of the Haar measure implies that for every net $(f_\alpha)_{\alpha\in A}$ with the properties $$ \alpha\le\beta\quad\Rightarrow\quad \mathrm{supp} (f_\alpha)\supset \mathrm{supp} (f_\beta) $$ and $$ \bigcap_{\alpha\in A}\mathrm{supp}(f_\alpha)=\{1\}, $$ and any two $\phi,\psi\in C_c^+(G)$, the quotient $ \frac{(\phi,f_\alpha)}{(\psi,f_\alpha)} $ converges to $\frac{\int_G\phi\,d\mu}{\int_G\psi\,d\mu}$. My question is this: For given $\phi\in C_c^+(G)$, does the quotient $$ \frac{[\phi,f_\alpha]}{(\phi,f_\alpha)} $$ converge to 1?

In case this is not true for any net as above, does there exist one net with this property?

It would be enough to assume that $G$ is first countable, so instead of nets you may as well use sequences.