Here is the question: Let $(\mathbb{R}, S, \lambda)$ be the measure space. Let $\lambda$ be denoted as the Lebesgue measure. Show that for every lebesgue integrable function, $f:\Bbb R\to\Bbb R$ and $\forall\varepsilon > 0$, there exists a continuous lebesgue integrable function $g:\Bbb R\to\Bbb R$ s. t. $\int_{\Bbb R} |f-g| d\lambda < \varepsilon$.
To be honest, I do not know what to do in this problem. Here is what I have so far: I am defining the sets as follows: $L = \{ f: f \text{ is Lebesgue integrable} \}$ and $G = \{ g \in L : \text{ there exists a continuous function } h \in L \text{ such that} \int_{\Bbb R} |g -h| d\lambda < \varepsilon \}$. Then I want to show that $L = G$. Is this the way to go here? Otherwise, what other way should I approach this problem?
Thank you for your help!!