so in looking at this one qual... part a was this one, part b was show what the limit as $s\rightarrow\infty$. ($f\in L^1$) For both the answer is 0; I get that, but I'm trying to insure the proof would be as 'clean' as possible.
For b, one should show that f can be approximated by functions with a compact closure; and for a, I believe one should show f can be approximated by a summation of characteristic functions and being tricky with the Lebesgue density theorem.
The problem I am hitting is that
$\int|f(t+s)-f(t)|dt\leq$
$\int|f(t+s)-\sum_na_n\chi_{E_n}(t+s)-(f(t)-\sum_na_n\chi_{E_n}(t))|dt+\\ \int|\sum_na_n(\chi_{E_n}(t+s)-\chi_{E_n}(t))|dt\leq \\ \int|f(t+s)-\sum_na_n\chi_{E_n}(t+s)|dt+\int|f(t)-\sum_na_n\chi_{E_n}(t)|dt+\\ \sum\int|a_n\chi_{E_n}(t)-\chi_{E_n}(t+s)|dt$
for the last term in the last inequality, I would need the integral to uniformly converge to 0 no matter the n value. (I am letting $E_n$ be defined by the distributional function of $f$). ... I guess this is the point where I end up believing that maybe my initial approach may have not been the appropriate approach. Is it ok to ask for a little bit of guidance here?
Here is a general approach to proving lots of things about the integral (I'm assuming the domain of integration is $\mathbf R$... if not make the obvious changes to notation):
First show the result is true for $f = \chi_{(a,b)}$.
Second show that if $\alpha \in \mathbf R$, then the result is true for $\alpha f$ whenever it is true for $f$.
Third show that if the result is true for functions $f_1,\ldots,f_n$ then it is true for their sum.
Fourth show that if the result is true for a sequence of functions $\{f_n\}$ and $f_n \to f$ in $L^1$, then the result is true for $f$.
Using these steps you can prove the result is true for the following types of functions: \begin{align*}\text{indicator functions of intervals} &\rightarrow \text{indicator functions of open sets with finite measure} \\ &\rightarrow \text{indicator functions of measurable sets with finite measure} \\ &\rightarrow \text{simple functions} \\&\rightarrow L^1\text{ functions}\end{align*}