I have a double sequence of probability density functions $\rho_{n,m}\in C^\infty(\mathbb R^d$). Suppose that for every test function $\varphi\in C_c^{\infty}(\mathbb R^d)$
$\lim_{m\to\infty}\lim_{n\to\infty}\int\varphi(x)\rho_{n,m}(x)dx = \int \varphi(x) \nu(x)dx$;
$\lim_{n\to\infty}\lim_{m\to\infty}\int\varphi(x)\rho_{n,m}(x)dx = \int \varphi(x) \mu(x)dx$
where $\mu,\nu\in C^{\infty}(\mathbb R^d)$ are suitable probability density functions.
Now fix $\psi\in C^\infty(\mathbb R^d)$, $\psi\geq0$ such that
$$\int\psi(x)\nu(x)dx<\infty \quad,\quad \int\psi(x)\mu(x)dx<\infty \;.$$
Can I conclude that there exist $C\in\mathbb R$, $\bar n,\bar m\in\mathbb N$ such that $$ \int \psi(x)\rho_{n,m}(x)dx \,\leq\, C $$ for all $n>\bar n\,$, $m>\bar m\,$?
If this is not true in general, what could be a possible strategy to deal with a particular case?