$f_n(x) = \chi_{B_{\frac{1}{n}}(x_0)}/\lambda^N({B_{\frac{1}{n}}(x_0)}) \longrightarrow \delta_{x_0}$ in $\mathscr{D'}(\Omega)$

Question

Given $x_0 \in \Omega$ and $f_n(x) = \chi_{B_{\frac{1}{n}}(x_0)}/\lambda^N({B_{\frac{1}{n}}(x_0)})$, where $\lambda^N$ is the N-dimensional Lebesgue measure, show that the distribution $T_{f_n} \longrightarrow \delta_{x_0}$ in $\mathscr{D'}(\Omega)$, where $T_{f_n}= <f_n,\phi>=\int_{\Omega}f_n\phi d\mu$.

I know that a sequence of distributions $\{T_{f_n}\}_{n\in \mathbb{N}}$ converges to $T \in \mathscr{D'}(\Omega) $ if $<f_n,\phi> \longrightarrow <T,\phi> \forall \phi \in \mathscr{D}(\Omega) $.

So, in this case I have to evaluate if $<f_n,\phi> \longrightarrow <\delta(x_0),\phi> $ in $\mathscr{D'}(\Omega)$ $$\int_{\Omega} \frac{1}{\lambda^N({B_{\frac{1}{n}}(x_0)})}\chi_{B_{\frac{1}{n}}(x_0)} \phi(x) dx \to \phi(x_0) $$

But I don't know how to proceed from here. Have you any suggestions?

Thanks in advance.

Answer 1

I am pretty sure this is a homework question, so I will just leave an outline and you will need to fill in the gaps.

Assuming your test function $\phi$ is locally Lipschitz (which will be the case if it is continuously differentiable), then $\|\phi(x)-\phi(x_0)\|<C(\|x-x_0\|)$ for all $x\in B(x_0,1/n)$ with $n$ sufficiently large. Now combine the triangle inequality for integrals with this bound to prove the limit you are after.