I want to find a sequence $(f_n) \subset \mathcal L^1(\mathbb R)$ of integrable functions which uniformly converges to $0$ but with $\int f_n \nrightarrow 0$. I came up with the following example: the density functions for a sequence of normal random varialbes with mean $0$ and variance $n$. That is, $$f_n := \frac{1}{\sqrt{2\pi n}} e^{-\frac{x^2}{2n}}.$$
Since its probability density, we have $\int f_n = 1$ for all $n$. Could anyone point out whether it is a valid example, please? Thank you!
Yes it is, as you already pointed out $ f_n \in \mathcal{L}^1(\mathbb{R}) $ with $ \int f_n = 1 $. Moreover, $ \| f_n \|_{\infty} = \frac{1}{\sqrt{2 \pi n}} $, so the $ f_n $'s converge uniformly to $ 0 $.
Another example would be a family of "streched" identicator functions: take $ f_n = \frac{1}{n} \mathbb{1}_{[0,n]} $, then $ \int f_n = 1 $ and $ \| f_n \|_{\infty} = \frac{1}{n} $, so the $ f_n $'s also converge uniformly to $ 0 $. You could also take similar functions, e.g. smooth bump functions etc.