Let $E ⊂ [0,1]$ be a measurable set, $m(E) ≥ \frac{99}{100} .$ Prove that there exists $x ∈ [0,1]$

Question

I need some help on the following real analysis past qual problem. I would appreciate some help.

Let $E ⊂ [0,1]$ be a measurable set, $m(E) ≥ \frac{99}{100} .$ Prove that there exists $x ∈ [0,1]$ such that for any $r ∈ (0, 1),$ $m(E ∩ (x − r, x + r)) ≥ \frac{r}{4} .$

Is there a way to use the Hardy-Littlewood maximal inequality?

Answer 1

I think this is just about compactness. Assume that for every $x \in [0,1]$, there is an interval $I$ about $x$ such that

$$m(E\cap I) < \frac{1}{8} m(I)$$ then there will be a finite subset of these intervals which cover the unit interval. Now by setting these intervals one after the other they can be chosen so that any interval overlaps only with following and the the previous intervals. This implies that $$m(E) \leq \frac{1}{8} \sum m(I) \leq \frac{1}{8} 2 m([0,1])=\frac{1}{4}$$