I know that for showing precompactness of subsets of continuous functions, Arzelà-Ascoli is the tool of choice. However, the setting I'm facing here are functions in $L^1$.
More precisely:
Let $$ \begin{align} E_1 &= \{ f:(0,1)\to\mathbb{R}:\,f(x) = x^{-\alpha},\,0\leq\alpha < 1\}\\ E_2 &= \{ f:(0,1)\to\mathbb{R}:\,f(x)=x^{-\alpha},\,-\infty < \alpha\leq 1-\delta\}\text{ with fixed } \delta > 0\\ E_3 &= \{ f:(0,1)\to\mathbb{R}:\,f(x) = \sin(\omega x),\,\omega\in\mathbb{R}\} \end{align} $$ Show whether those sets are bounded and/or precompact as subsets of $L^1((0,1))$.
Boundedness is clear, we can just calculate the anti-derivative for functions from $E_1$ and $E_2$ and see that they need to be bounded for functions from $E_2$ but not for functions from $E_1$.
For $E_3$ just estimate $\sin(x) \leq 1$ and you see that $E_3$ is also bounded.
Now how to show precompactness? I need to show that for all $\varepsilon>0$ there is a finite covering of $E_1,E_2,E_3$ using $\varepsilon$-balls (or not).
In my opinion we need to somehow select a finite family of functions from $E_i$, say $f^i_1,\dots,f^i_k$ s.t. all $f^i\in E_i$ belong to some ball $$B(f_j^i,\varepsilon) = \{g\in E_i:\, \|f_j^i - g\|_{L^1} < \varepsilon\}$$
I don't understand how to choose such a set. My intuition for this is: $$\text{bounded} \Leftrightarrow \text{Precompact}$$ which is obviously wrong.
Maybe someone can show me how to start this.
Thanks a lot!