If $f(x)$ is nonnegative function satisfying $\int_{-\infty}^{\infty} f(x)dx =1$ show that ${mf(mx)}$ is delta sequence.
We need to check $ lim \int_{-\infty}^{\infty} {mf(mx)} f(x)dx $ = $f(0)$ as $m$ goes to $\infty $. But I can not. Please, help me ?
Take $\phi \in C_c^\infty(\mathbb{R})$. Then $$ \int_{-\infty}^{\infty} m \, f(mx) \, \phi(x) \, dx = \{ x' = mx \} = \int_{-\infty}^{\infty} f(x') \, \phi(x'/m) \, dx' \\ \to \int_{-\infty}^{\infty} f(x') \, \phi(0) \, dx' = \int_{-\infty}^{\infty} f(x') \, dx' \, \phi(0) = \phi(0), $$ where the Dominated Convergence Theorem has been used when taking the limit.