Show that $\lim_{t \to 0} \int_{\mathbb{R}^d}|f(x)-f(x-t)|dx = 0$ Let $f \in L^1(\mathbb{R}^d)$. Show that $\lim_{t \to 0} \int_{\mathbb{R}^d}|f(x)-f(x-t)|dx = 0$.
What I want to do is bound $|f(x)-f(x-t)|$ above by something and then use the Lebesgue Dominated Convergence theorem to move the limit inside the integral sign, which would solve the problem. I know that $f$ is finite for almost every $x \in \mathbb{R}^d$, but I don't know that it's bounded, so I'm not sure if this is a feasible approach. In a previous question on this homework set I showed that $\int_{\mathbb{R}^d} f(x)dx = \int_{\mathbb{R}^d} f(x+a)dx$, for all $a \in \mathbb{R}^d$, but I'm not sure if that can be applied to this problem. Any hints will be appreciated (it is a homework problem, so I would prefer hints over a complete solution).
 A: First of all, I assume you want to ask 
$$ \lim_{t\to 0}\int|{f(x)-f(x-t)}|\,dx=0$$
but not
$$ \lim_{t\to \infty}\int|{f(x)-f(x-t)}|\,dx=0$$
You want $t\to0$ but not $t\to\infty$ right?
Next, from your previous hw, we know
$$ \int_{R^N} f(x)\,dx = \int_{R^N} f(x-t)\,dx$$
Also, I assume you know that the continuous function is dense in $L^1(R^N)$ and hence we can have a sequence of functions $(f_n)\subset C_c^0(R^N)$ such that $f_n\to f$ in $L^1$.
Now here is how LDCT comes in. We have $|f_n(x)-f_n(x-t)|\leq |f_n(x)|+|f_n(x-t)|$ by triangle inequality for a.e. $x\in R^N$.  Hence we could dominate $|f_n(x)-f_n(x-t)|$ by $|f_n(x)|+|f_n(x-t)|$. The later item is integrable because $f_n\to f$ in $L^1$ and hence we have $\|f_n\|_{L^1}\leq \|f\|_{L^1}+1$ for $n$ large enough. Now, LDCT will tell you that you could move your limit from outside into inside and you have
$$ \lim_{t\to 0}\int|{f_n(x)-f_n(x-t)}|\,dx=0$$
To conclude, we write
$$ \int|{f(x)-f(x-t)}|\,dx\leq  \int|{f(x)-f_n(x)}|\,dx +\int|{f_n(x)-f_n(x-t)}|\,dx+ \int|{f(x-t)-f_n(x-t)}|\,dx $$
Notice that for any given $\epsilon>0$, you could choose $n$ large enough so that the first and last term is less then $\epsilon/3$. Next take $t$ small enough you have second term less the n$\epsilon/3$. Sum up, you have whole thing is less then $\epsilon$ and hence you could conclude your result.
The first and last item goes to $0$ because $f_n\to f$ in $L^1$.
If you don't know why $C_c^0$ is dense in $L^1$, actually $L^p$ for $1\leq p<\infty$, I think you could find prove in Rudin's book.
The key is LDCT only require the dominate function to be integrable, but not finite nor bounded.
