Prove that a relatively compact subset of $L^p$ is bounded.

Question

Let $p\in [1,\infty)$, $A\subset L^p(\mathbb R^m)$ relatively compact and $\lambda^m$ be the Lebesgue measure on $\mathbb R^m$. Prove:

a) $A$ is bounded.

b) $\lim_{y \to 0}\sup_{f \in A} \int_{\mathbb R^m} |f(x+y)-f(x)|^p d\lambda^m(x)=0$

c) $\lim_{r \to \infty}\sup_{f \in A} \int_{|x|>r} |f(x)|^p d\lambda^m(x)=0$.

I can't get any further:
a): Every relatively compact set in a normed space is totally bounded, which implies boundedness. (Right?)
b), c): Cover A by balls and approximate the centers by simple functions. (How do I do this formally?)

Answer 1

We can show that b) and c) hold when $A$ is replace by $B$:

1. if $B=\{\chi_S\}$ where $S\subset\mathbb R^n$ is an open set of finite measure;

2. when $B=\{\chi_S\}$ where $S$ is any measurable subset of $\mathbb R^n$ of finite measure;

3. when $B=\{f_0\}$, where $f_0\in\mathbb L^p$;

4. when $B$ is finite.

Then conclude by a $2\varepsilon$-argument.