Approximate an integrable function using a simple function (Proving existance)

Question

Let $f \in L^1(\mathbb{R})$, and let $\epsilon > 0$. Show that exists simple function $g=\sum_{k=1}^{n}c_k 1_{A_k}$, such that, $$\int_\mathbb{R} |f(x)-g(x)|dx \leq \epsilon$$,and such that $n \in \mathbb{N}$ and $A_k$ are bounded. I can't find a good partition, and I don't understand how to find the $n$ which will satisfy my needs. It's like I miss information.

Answer 1

By definition of Lebesgue integral, we have for a non-negative integrable function $g$ and any positive $\varepsilon$ a simple function $h$ such that $\int_{\mathbb R}|g-h|\mathrm d\lambda\lt\varepsilon$. By splitting the function on its negative and positive part, we can extend this to any integrable function.

It means that we can find an integer $n$, real numbers $c_1,\dots,c_n$ and a collection of disjoints set with finite measure such that $$\int_{\mathbb R}\left|f-\sum_{j=1}^nc_j\mathbf 1(A_j)\right|\mathrm d\lambda\lt \varepsilon/2.$$ In general, the sets $A_j$ may not be bounded. Nevertheless, since $\lambda\left(A_j\cap[-l,l]\right)\to \lambda(A_j)$ as $l$ goes to infinity, we can take for each $j\in\{1,\dots,n\}$ an integer $n_j$ such that $|c_j|\left(\lambda(A_j)-\lambda\left(A_j\cap[-n_j,n_j]\right)\right)\leqslant \varepsilon/(2n).$