limit question on Lebesgue functions Let $f\in L^1(\mathbb{R})$.  Compute $\lim_{|h|\rightarrow\infty}\int_{-\infty}^\infty |f(x+h)+f(x)|dx$
If $f\in C_c(\mathbb{R})$ I got the limit to be $\int_{-\infty}^\infty |f(x)|dx$.  I am not sure if this is right.

*

*Let $f$ a continuous function with compact support, say contained in $-[R,R]$. For $h\geq 2R$, the supports of $\tau_hf$ and $f$ are disjoint (they are respectively $[-R-h,R-h]$ and $[-R,R]$ hence 
\begin{align*}
\int_{\Bbb R}|f(x+h)+f(x)|dx&=\int_{[-R,R]}|f(x+h)+f(x)|+\int_{[-R-h,R-h]}|f(x+h)+f(x)|\\
&=\int_{[-R,R]}|f(x)|+\int_{[-R-h,R-h]}|f(x+h)|\\
&=2\int_{\Bbb R}|f(x)|dx.
\end{align*}

*If $f\in L^1$, let $\{f_n\}$ a sequence of continuous functions with compact support which converges to $f$ in $L^1$, for example $\lVert f-f_n\rVert_{L^1}\leq n^{—1}$. Let $L(f,h):=\int_{\Bbb R}|f(x+h)+f(x)|dx$. We have
\begin{align}
\left|L(f,h)-L(f_n,h)\right|&\leq 
\int_{\Bbb R}|f(x+h)-f_n(x+h)+f(x)-f_n(x)|dx\\
&\leq \int_{\Bbb R}(|f(x+h)-f_n(x+h)|+|f(x)-f_n(x)|)dx\\
&\leq 2n^{-1},
\end{align}
and we deduce that 
$$|L(f,h)-2\lVert f\rVert_{L^1}|\leq 4n^{-1}+|L(f_n,h)-2\lVert f_n\rVert_{L^1}|.$$
We have for each integer $n$,
$$\limsup_{h\to +\infty}|L(f,h)-2\lVert f\rVert_{L^1}|\leq 4n^{—1}.$$
This gives the wanted result. 

