Proving a sequence of locally integrable functions goes to Dirac delta

Question

Attempt: I have done this type of questions before too and usually a substitution $x =\frac{x}{m}$ does the job. But it does not work in the case. How should I proceed?

Any hints will be appreciated!

Answer 1

For any test function $g$ we have $$|\int f_n(x)[g(x)-g(0)]dx|\leq \epsilon \int_{\{|x| \leq 1/n\}} |f_n(x)| dx$$ provide $n$ is so large that $$|g(x)-g(0)| <\epsilon$$ for $|x| \leq 1/n$. Hence $$|\int f_n(x)[g(x)-g(0)]dx|\leq C\epsilon$$ for $n$ large enough. We have proved that $$\int f_n g \to g(0)$$ for any test function $g$ (since $\int f_n(x)g(0)dx=g(0)\int f_n(x)dx=g(0)$).