From pointwise convergence to uniform convergence

Question

Consider a real-valued random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$.

Let $a\in \mathbb{R}$.

Let $\{f_n(X, a)\}_n$ be a sequence of real-valued random functions.

Consider a non-random function $f:\mathbb{R}\rightarrow \mathbb{R}$ continuous on the entire domain and such that $\int_{-\infty}^\infty |f(a)|da<\infty$

Assume $\lim_{n\rightarrow \infty} E_{\mathbb{P}}(f_n(X,a))-f(a)=0$ $\forall a\in \mathbb{R}$

Could you help me to show that all these assumptions imply $\lim_{n\rightarrow \infty}\sup_{a} |E_{\mathbb{P}}(f_n(X,a))-f(a)|=0$?

Answer 1

This may be not true: take $f_n(X,a):=f(a)+a/n$: the assumptions are satisfied but $\sup_{a} |E_{\mathbb{P}}(f_n(X,a))-f(a)|$ is infinite.