This problem is from S.-T. Yau College Student Mathematics Contests 2018. I have som basic knowledge of Lebesgue measures, but I have not systematically studied Borel measures. I don't have any idea about this problem.
Let $\mu$ be a Borel measure on $\mathbb{R}^n$. Let $\rho$>0, a fixed positive number, and $B_{\rho}(x)=\left\{y\in\mathbb{R}^n|d(x,y)<\rho\right\}$. For $x\in\mathbb{R}^n$, denite a function:$$\theta(x):x\rightarrow\mu(\overline{B_{\rho}(x)})$$
- Show that $\theta$ is upper semi-continuous, i.e. for every $x\in\mathbb{R}^n,\ \theta(x)\ge\limsup_{y\rightarrow x}\theta(y)$.
- Give an example of a Borel measure $\mu$, such that the function $\theta$ is not continuous.