Let $A\subset \mathbb{R}^n$ be a bounded set and $\alpha \in (0,1)$ such that: $\mu^*(A\cap Q)≤\alpha\cdot\mu(Q)$. Show that A is the null set. 
Let $A\subset \mathbb{R}^n$ be a bounded set and $\alpha \in (0,1)$ such that for all cuboids $Q\subset\mathbb{R}^n$: $$\mu^*(A\cap Q)≤\alpha\cdot\mu(Q)$$ Show that A is the null set.

How to begin? What conclusions can I draw from A being bounded?

Let $\beta< 1 $ a bit larger than $\alpha$.
Let's first agree on this: for every box $Q$ there exists a countable family of boxes $(Q_i)$ inside $Q$ that cover $Q\cap A$ and such that 
$$\sum \mu(Q_i) \le \beta\cdot \mu(Q)$$
Repeat this procedure several times ($N$ times) to get a countable family of boxes covering $Q\cap A$ with sum of volumes $\le \beta^N \cdot \mu(Q)$.  
