I have to prove that if $f\in L^p(\mathbb{R}^n)$, $1\leq p<\infty$, then $f_h(x)=f(x+h)$ satisfy $||f_h-f||_p\to 0$ if $h\to 0$. I have the idea but I don't finish writing it in detail and formally. Let's see if you can help me.
We choose a succession $\{ g_k\}_k \in C_c(\mathbb{R}^n)$ such that $||f-g_k||_p\to 0$. Then,
\begin{align*} ||f(x+h)-f(x)||_p &\leq ||(f-g_k)(x+h)||_p+||g_k(x+h)-g_k(x)||_p+||g_k(x)-f(x)||_p\\ &= 2||f-g_k||_p+||g_k(x+h)-g_k(x)||_p. \end{align*}
In the last step, we have applied the change $y=x+h$ in the first term.
Since $g_k$ functions are continuous with compact support, applying the dominated convergence theorem, it is shown that $||g_k(x+h)-g_k(x)||_p\to 0$ if $h\to 0$.
I want to use it to prove the following result: let $\phi \in C_c^\infty (\mathbb{R}^n)$, that is, a class $C^\infty$ function with compact support such that $\int_{\mathbb{R}^n}\phi =1$. For $t>0$, let's consider the function $\phi_t(x)=t^{-n}\phi (x/t)$. Then the function $f_t=\phi_t *f$ is a class function $C^\infty (\mathbb{R}^n)$ that fulfills $||f_t||_1\leq ||f||_1$ and $||\phi_t * f-f||_1\to 0$ when $t\to 0^+$.