If $\sup_{x \in \mathbb{R}^n} |f_m(x)|\rightarrow 0$ for $f$ a Schwartz function (i.e. $f$ is smooth and decays rapidly), is it true that $\|f_m\|_{L^1}=\int_{\mathbb{R}^n} |f_m(x)| dx \rightarrow 0$ as $m \rightarrow \infty$?
This seems to be true but I am not sure how to justify it.
Not true. Pick $f_m = \frac{1}{m} 1_{\lbrack 0, m \rbrack }$.