Given:
Let $f(x)$ be a bandlimited density with the band limit $\alpha$, that is, a density with a finite support of its Fourier transform, centered at the origin of the real axis and of length $2\alpha$.
Let $\{f_n(x)\}$ be a sequence of bandlimited densities with the band limit $\alpha$ that uniformly converges to $f(x)$ almost surely, that is, $$ f_n(x) \xrightarrow[]{\text{a.s.}} f(x), \quad \text{uniformly},$$ or equivalently $$ \Pr\left( \lim_{n\rightarrow\infty} \sup_{x\in\mathbb{R}} | f_n(x) - f(x) | = 0 \right) = 1 . $$
Let $\psi: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ be a continuous mapping.
I am ready to have mild additional technical assumptions on $f(x)$, $\{f_n(x)\}$, and $\psi$.
To be proven:
I would like to prove that $$ \int_{\mathbb{R}} \psi\left(f_n(x)\right) \, dx \xrightarrow[]{\text{a.s.}} \int_{\mathbb{R}} \psi\left(f(x)\right) \, dx ,$$ or equivalently $$ \Pr\left( \lim_{n\rightarrow \infty} \int_{\mathbb{R}} \psi\left(f_n(x)\right) \, dx = \int_{\mathbb{R}} \psi\left(f(x)\right) \, dx \right) = 1.$$ Any idea on how to approach this problem?