I have a question about Riesz Kolmogorov Frechet.
$X \subset \mathbb{R}^n$ measurable and $1\leq p<\infty$. A set $A\subset L^p(X)$ is precompact if and only if
$ sup_{f\in A} ||f||_{L^p(X)} < \infty$
$sup_{f\in A} ||\overline{{ f}}(\cdot+h)-\overline{{ f}}||_{L^p(X \cup (X-h))}\rightarrow0$ for $|h|\rightarrow 0$
$sup_{f\in A} ||f||_{L^p(X/B_R(0))}\rightarrow 0$ for $R \rightarrow 0$
where $\overline{ f}$ is the trivial $0$ extension on $\mathbb{R}^n$
The difference i spot is that its $L^p(X)$ instead of $L^p(\mathbb{R}^n)$ and $\overline{{ f}}$ instead of $f$. I dont see how to prove this though. Can anyone help me?