Let $\mu$ be an outer measure on $X$. Let $A \subset X$. $x \in X$.
Let $B_\rho(x)$ be the ball of radius $\rho$ centered at $x$.
I read that if all Borel sets are $\mu$-measurable, and $\mu(B_\rho(x)) < \infty$, then
$$ \mu(A \cap B_\rho(x)) \geq \limsup_{y \to x} \mu(A \cap B_\rho(y)).$$
Why is this true?