How to prove the existence of continous functions who approximates Lebesgue integrable?

Question

I am trying to prove the Lebesgue Riemann lemma but before I need to prove this:

Let be $\left(\mathbb{R},A^*_{\mathbb{R}},\overline{\lambda}\right)$ the measurable Lebesgue space of $\mathbb{R},$ $p\in[1,\infty), \epsilon>0$ and $|f|^p$ integrable. Then exists $A>0$ and a continuous function $g$ s.t. $g(\mathbb{R}\setminus[-A,A])\subseteq\{0\}$ and $\int|f-g|^p d\overline{\lambda}<\epsilon.$