Sequence of continuous functions that convergent to the indicator function pointwisely.

Question

Let $L=[a_1, b_1]\times \cdots \times [a_n,b_n]$ be a closed rectangle in $\mathbb{R^n}$.

Prove that there exists a sequence of continuous functions $\{ f_n \}_{n=1}^{\infty}$ s.t. each $f_n$ is defined on $\mathbb{R^n}$ and pointwisely convergents to the indicator function $\chi_L$.

I thought $f_n=\chi_{[a_1-\frac{1}{n}, b_1+\frac{1}{n}]\times \cdots \times [a_n-\frac{1}{n},b_n+\frac{1}{n}]}$ seems to work but this is not continuous.

I have no idea how I should determine $f_n$.

Thank you for your help.

Answer 1

You can convolve the indicator function with the $n$-dimensional heat kernel

$$\frac{1}{(4\pi t)^{n/2}}\int_L \mathrm{e}^{-\dfrac{\|\mathbf{x}-\mathbf{y}\|^2}{4t}} \mathrm{d}y $$

Taking $t=1/n$ in the above given an appropriate sequence $f_n$.