$L_p$ Spaces and limits of translated functions If $g\in L^p(\mathbb{R}^n)$  and $1\leq p<\infty$ then show $$\lim_{|t|\to \infty}\lVert g_{(t)}+g\rVert_p=2^{1/p}\lVert g\rVert_p,$$
where $g_{(t)}(x):=g(t+x)$.
Any hints? Try to give me only hints/outlines not complete solutions 
Not sure where to go from there?
 A: Hint: Prove this for compactly supported functions.
A: Hint: 


*

*First show it for characteristic functions $\chi_I$  where $I$ is some interval.

*Using 1. prove this for simple functions (i.e. finite sums $\sum \alpha_I\chi_I$).

*Prove the general statement by approximating a general function $g\in L^p$ by those of 2.
A: Hint:
Let's introduce $B_r(P)$ as it is defined and $\|\ f\|_{L_{p}{\{\text{Area of integrating}\}}}=\left(\int\limits_{\text{Area of integrating}}|f(x)|^p dx\right)^{\frac{1}{p}}$. The key to solving the problem are triangle inequalities as it is shown below:
$$\|g\|_{L_{p}\{B_{0.5|t|}(0)\}}\leqslant\|g_{(t)}+g\|_{L_{p}\{B_{0.5|t|}(0)\}}+\|g_{(t)}\|_{L_{p}\{B_{0.5|t|}(0)\}}$$
and:
$$\|g_{(t)}+g\|_{L_{p}\{B_{0.5|t|}(0)\}}\leqslant \|g\|_{L_{p}\{B_{0.5|t|}(0)\}}+\|g_{(t)}\|_{L_{p}\{B_{0.5|t|}(0)\}}$$
Notice that: $$\|g_{(t)}+g\|_{L_p}=\left(\|g_{(t)}+g\|_{L_{p}\{B_{0.5|t|}(0)\}}^p+\|g_{(t)}+g\|_{L_{p}\{B_{0.5|t|}(-t)\}}^p+\|g_{(t)}+g\|_{L_{p}\{\overline{B_{0.5|t|}(0)\cup B_{0.5|t|}(-t)}\}}^p\right)^{\frac{1}{p}}$$
