Let $G$ be a topological group and $U$ be a compact neighbourhood of the idenitity element $e\in G$. Let $f,f_0\in C_c^+(G)$ with $f_0\neq 0$. Let $(f:\textbf{1}_U)$ be the following infimum where $\textbf{1}_U$ is the indicator function of $U$.
$$\inf\left\{\text{$\sum_{i=1}^m c_i \mid c_1,\ldots,c_m>0$, and $\exists\,t_1,\ldots,t_m\in G$ such that $\forall\,x\in G : f(x)\leq \sum_{i=1}^m c_i \textbf{1}_U (t_ix)$} \right\}$$
We define a map $I_U$ on $C_c^+(G)$ by $I_U(f)=\frac{(f:\textbf{1}_U)}{(f_0: \textbf{1}_U)}$. Is the below equation true?
$$\bigcap_{V}\overline{L_V}=\left\{\lim_{U\to \{e\}}I_U \right\}$$ where $L_V:=\{ I_U: \text{$U\subseteq V$ is a compact neighborhood of $e\in G$} \}$ and $V$ runs over all neighbourhood of $e$.
In fact we know the intersection $\bigcap_{V}\overline{L_V}$ is nonempty. Thanks!