I was working in the next exercise but really I don't know how to start.
Let $f\in\mathcal{L}_p(\mathbb{R},\mathcal{B},\lambda)$ (i.e., the $\mathcal{L}_p$ space with the $\sigma$-algebra of Borel and the lebesgue measure), $p\in[1,\infty)$ and $\varepsilon>0$.
i) Prove that there exist $A>0$ and $g:\mathbb{R}\to\mathbb{R}$ continuous such that $g(x)=0$ for all $x\notin [-A,A]$ and $||f-g||_p<\varepsilon$. ii) Prove that there exist $g$ and step function (or step function by parts or differentiable) such that $||f-g||_p<\varepsilon$.
My idea stars with the next theorem by N.N Luzin:
1.- A function $f:\mathbb{R}\to\mathbb{R}$ is Lebesgue measurable if and only if for all $\varepsilon>0$ there exist $E$ and open set with $\lambda(E)<\varepsilon$ and such that $f_{|\mathbb{R}\setminus E}:\mathbb{R}\setminus E\to\mathbb{R}$ is continuous.
2.- A function $f:\mathbb{R}\to\mathbb{R}$ is Lebesgue measurable if and only if for al $\varepsilon>0$ there exist $g_\varepsilon:\mathbb{R}\to\mathbb{R}$ continuous such that $$\lambda\left(\left\{x\in\mathbb{R}\mid f(x)\neq g_\varepsilon(x) \right\} \right)<\varepsilon$$
With this theorem, for the given $\varepsilon>0$ we can take, by the theorem, a function $g_\varepsilon:\mathbb{R}\to\mathbb{R}$ such that $f=g$ except on a set of measure less than $\varepsilon$. But from here, how can I do to obtain that $||f-g||_p<\varepsilon$? I don't know. I also thought about to try to aproximate $f$ by characteristic functions and in the limit obtain the result, but I don't know how. I really appreciate any help. Thanks!
Assume that $f = \chi_{E}$ where $E$ is of finite measure. By regularity of Lebesgue measure, there is an open set $O \supset E$ and a compact set $K \subset E$ such that $\lambda(O \setminus K) < \varepsilon$. By the $C^{\infty}$ Uryshon lemma, there is a $\phi \in C_c^{\infty}(\mathbb{R}^n)$ such that $\chi_{K} \leq \phi \leq \chi_{O}$. Then $\lVert \chi_{E} - \phi \rVert_{L^p} \leq \varepsilon^{1/p}$.
Thus $C_c^{\infty}(\mathbb{R}^n)$ can approximate any indicator of a set of finite measure, and therefore can approximate any function in $L^p(\mathbb{R}^n)$, that is, $C_c^\infty(\mathbb{R}^n)$ is dense in $L^p(\mathbb{R}^n)$.