I would like your opinion on a computation i found in a statistical mechanics paper :
Let $\nu$ a prob measure on $\mathbb{R}^{d}$, $V:\mathbb{R}^{d} \rightarrow \mathbb{R}$ continuous, belongs to $L^1(\nu)$, with $\lim \limits_{|x|\rightarrow\infty} V(x) = \infty$ and $\int \exp(-V(x))dx <\infty $ (maybe not necessary. i am not sure). Also, let $d\nu_n(x) = (\nu\ast \rho_n) (x)dx$ where and $(\rho_n)_n$ a sequence of mollifiers. I want to see if the following holds : $\sup \limits_{n} \int |V| d\nu_n <\infty $. (in case it doesn't hold, can I get the same result but with $V$ instead of $|V|$ in the integral? ) Could you give me some help ?
My approach so far : $\int |V|(x) d\nu_n(x) = \int |V|(x) (\nu\ast \rho_n) (x)dx = \int \limits_x |V|(x) \int\limits_y \rho_n(x-y)d\nu(y) dx = \\ \int \limits_x \int\limits_y |V|(x) \rho_n(x-y)d\nu(y) \, dx \overset{{\textit{Tonelli}}}{=} \int \limits_y \int\limits_x |V|(x) \rho_n(x-y) dx d\nu(y) = \int \limits_y (\rho_n\ast |V|)(y) d\nu(y) $ \