Let $f$ be a real valued function that is integrable on $\mathbb{R}$ and let $\varepsilon >0$. Show that there is a continuous function $g$ that is identically zero outside some interval and such that $$\displaystyle\int_\mathbb{R} |f-g| <\varepsilon.$$ (Hint: Lusin's theorem}
So I used Lusin's theorem and a function $g$ appears from Lusin's but I don't know how this function will satisfy the conclusion of the problem. Any help will be appreciated.