Smooth approximation identities

Question

Let $G$ be a Lie group. Let $\mathcal U$ be a neighbourhood base at $1$ of G. Does there exist $\{\psi_U:U\in\mathcal U\}$ with the following properties support of $\psi_U$ is compact and contained in $U,$ $\psi_U\geq 0,$ $\psi_U(x^{-1})=\psi_U(x)$, $\int_G\psi_U(x)dx=1$ and $\psi_U$ is smooth?

Answer 1

It seems so. Start with any bump function $\phi_U \geq 0$ around $1$ whose compact support is contained in $U$. Then let $\varphi_U(x) \doteq \phi_U(x)\phi_U(x^{-1})$. This is smooth, also has compact support contained in $U$, and the property $\varphi_U(x^{-1}) = \varphi_U(x)$ holds by construction. Then set $$\psi_U(x) = \frac{1}{\int_G \varphi_U(x)\,{\rm d}x} \varphi_U(x).$$