Definition

Let $x \in\mathbb{R}$. A sequence of Borel subsets $(E_{j})_{j\ge 1}$ is said to shrink nicely to $x$ if and only if $\exists\alpha>0$ with the following property: > There is a sequence of Open intervals $J_{j}$ with $\lim_{ j \to \infty }m(J_{j})=0$ such that $$E_{j}\subseteq J_{j}\text{ and }m(E_{j})\ge \alpha \cdot m(J_{j})\quad\forall j\ge 1$$