Suppose that $\lambda^{d}$ is the Lebesgue-Borel measure on $\mathbb{R}^d$ and, if $r>0$, $H\_r$ is the map on $\mathbb{R}^d$ defined by $H\_r(\overline{x})=r\overline{x}$. Furthermore, $K$ is a non-negative real function over $\mathbb{R}^d$ such that $\int\_{\mathbb{R}^d}Kd\lambda^{d}=1$. If $f$ is a continuous real function over $\mathbb{R}^d$ with compact support, define $I\_r(f)=\int\_{\mathbb{R}^d}f\circ H\_r^{-1}.Kd\lambda^{d}=\int\_{\mathbb{R}^d}f(\dfrac{\overline{x}}{r})Kd\lambda^{d}$. Prove that $\lim\_{r\rightarrow \infty}I\_r(f)=f(\overline{0})$.
I can prove (using dominated convergence) that the sequence $\left ( I\_n(f) \right )\_{n \in \mathbb{N}}$ converges to $f(\overline{0})$, but how go from there?