Let $(X,d)$ be a metric space and $\mu:2^X\to[0,\infty]$ be an exterior measure on $X$, $\mathcal{M}:=\{E\subset X:\forall S\subset X\ \ \ \mu(S)=\mu(S\cap E)+\mu(S-E)\}$. Suppose $\mathcal{B}(X)\subset\mathcal{M}$.
Then Leon Simon Lectures on geometric measure theory chapter 1 section 3 remark 3.1 states that \begin{eqnarray} \mu(B(x;\rho))\geq\limsup_{y\to x}\mu(B(y;\rho)) \end{eqnarray} , where $x\in X$, $\rho>0, $ $B(x;\rho):=\{a\in X:d(x,a)\leq\rho\}$.
I have no idea to prove this statement. Please tell me any idea. Thank you.