Let $f_\epsilon\in L^1(\mathbb{R}^n)$ be a function which depends on a parameter $\epsilon\in(0,1)$, and is such that
- $\operatorname{supp}{f_\epsilon}\subset\{|x|\leq\epsilon\}$,
- the total integral of $f_\epsilon$ is $1$, and
- $\displaystyle\int_{\mathbb{R}^n}{|f_\epsilon(x)|\,dx}\leq\mu \lt \infty$ for $\epsilon\in(0,1)$.
How do I show that $f_\epsilon\rightarrow\delta$ (in the space of tempered distributions on $\mathbb{R}^n$) as $\epsilon\rightarrow0^+$,
i.e. how do I show $\int_{\mathbb{R}^n}{f_\epsilon(x)\,\phi(x)\,dx}=\int_{|x|\leq\epsilon}{f_\epsilon(x)\,\phi(x)\,dx}\;\xrightarrow{\varepsilon \to 0^+}\;\phi(0)$ for all test functions $\phi$?