Smooth approximation of characteristic function of a bounded open set

Question

Let $U$ be an open bounded set of $\mathbb{R}^n$. Is it possible to approximate $\chi_U$ as almost everywhere limit of increasing sequence of smooth functions?

Answer 1

Let $$f_n(x)=\frac{n}{\sqrt{\pi}}e^{-(nx)^2}$$ which has integral $1$ and approaches the dirac delta function as $n\to \infty$. Then the convolution $\chi_U*f_n$ is smooth for each $n$ as $f_n$ is smooth and we have $$\frac{d^k}{dx^k}\int_{\mathbb R}\chi_U(t)f_n(x-t)dt=\int_{\mathbb R}\chi_U(t)\frac{d^kf_n}{dx^k}(x-t)dt$$ and converges everywhere to $\chi_U$.