Lebesgue integral converging to $L^1(\mathbb{R})$ norm I'm working on the following integral: Given $f\in L^1(\mathbb{R})$ $$\lim_{x \to 0}\int\limits_{-\infty}^\infty |f(x+t)+f(t)|\,dt=2||f||_1$$
I'm still learning the techniques for dealing with integrals such as these. I've tried using dominated convergence with the characteristic function, but the details are still a little murky as I haven't seen many problems like this before.
 A: Idea. In this problem, note that the family of non-linear functionals $(T_x)_{x>0}$ defined by
$$ T_x(f) = \int_{\mathbb{R}} \left| f(x+t) + f(t) \right| \, \mathrm{d}t $$
is uniformly Lipschitz, in the sense that
$$ \left| T_x (f) - T_x (g) \right| \leq 2\| f - g\|_1 $$
for all $f, g \in L^1(\mathbb{R})$ and $x > 0$. Its implication to this problem is that, we can first show the claim for nice functions and then extend it to all of $L^1(\mathbb{R})$ by the continuity.
Sketch of Proof. Let $g \in C_c(\mathbb{R})$ be a compactly supported continuous function on $\mathbb{R}$. Then the uniform continuity of $g$ implies that $g(\cdot + x) \to g(\cdot)$ uniformly on $\mathbb{R}$ as $x \to 0$. Together with the compactly-supportedness, we get
$$ \lim_{x \to 0} T_x(g) = 2\|g\|_1. $$
Now let $f \in L^1(\mathbb{R})$ and $g \in C_c(\mathbb{R})$ be arbitrary. Then
\begin{align*}
\left| T_x(f) - 2\|f\|_1 \right|
&\leq \left| T_x(f) - T_x(g) \right| + \left| T_x(g) - 2\|g\|_1 \right| + \left| 2\|g\|_1 - 2\|f\|_1 \right| \\
&\leq \left| T_x(g) - 2\|g\|_1 \right| + 4\|f - g\|_1.
\end{align*}
So taking $\limsup$ as $x \to 0$ and using the above observation, we obtain
\begin{align*}
\limsup_{x\to 0} \left| T_x(f) - 2\|f\|_1 \right| \leq 4\|f - g\|_1.
\end{align*}
Since the left-hand side is independent of $g$ and $C_c(\mathbb{R})$ is dense in $L^1(\mathbb{R})$, the claim follows by taking $g \to f$ in $L^1(\mathbb{R})$.
