A sequence of test functions that converges to a charscteristic function from beiow.

Question

Let $E$ be a Borel bounded set of $\mathbb{R}^{n}$ and $\chi$ be the characteristic function of $E$. How would you construct a sequence $ \chi_{n}$ of nonnegative test functions bounded above by $\chi$ that tends to $\chi$, as $n\to\infty$?

Answer 1

Let $E$ be the set of all irrational numbers in $(0,1)$. If $f_n$'s are test functions such that $0\leq f_n \leq I_E$ then $f_n(x)=0$ for all rational numbers $x$ in $(0,1)$. Since test functions are continuous this implies $f_n\equiv 0$ on $(0,1)$. I no sense does $(f_n)$ converge to $I_E$.