$L_p$ Spaces and limits of translated functions

Question

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?

Answer 1

Hint: Prove this for compactly supported functions.

Answer 2

Hint:

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

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

3. Prove the general statement by approximating a general function $g\in L^p$ by those of 2.