So, I've recently studied Brezis' Functional Analysis book. In Chapter 4, we have the following:
Theorem 4.26 - (Kolmogorov, Riesz.) Let $\mathcal{F}$ be a bounded set in $L^p(\mathbb{R}^N)$, where $1\le p < \infty.$ Assume that $$\lim_{|h|\to 0} \|f(\cdot + h) - f(\cdot)\|_p = 0 \text{ uniformly in }f.$$ Then the closure of $\mathcal{F}|_{\Omega}$ in $L^p(\Omega)$ is compact for any measurable $\Omega\subset \mathbb{R}^N$ such that $|\Omega|<\infty$.
So I've gone through the proof and I believe I have understood it well - but to be honest, I cannot think of any concrete examples where I can apply the theorem.
What are some examples of bounded sequences $\{f_n\}\subset L^p(\mathbb{R}^N)$ that satisfy the uniform $L^p$-continuity condition? Some examples of bounded sequences $\{f_n\}\subset L^p(\mathbb{R}^N)$ where the $L^p$-continuity is not uniform in $n$?
I apologize if I seem to be "lacking in work," but I really cannot think of any solid examples by myself at the moment. Thanks in advance.