Show that the measure is equal to zero

Question

Let $\mu$ be a Borel measure in $\mathbb{R}$ such that $\mu(I) \leq v^a(I)$ for each bounded interval $I$, where $a>1$. Show that $\mu=0$.

($v(R)$ is the volume of $R$)

Do we maybe use the following to show that?

For each rectangle $R$, $m^*(R)=v(R)$.

Or do I have to do it otherwise?? Could you give me some hints??

Answer 1

For every interval $I =[x,y)$ and every $n \in \mathbb{N}$ we have that $$I= [x,y)= \bigcup_{k = 0}^{n-1} \underbrace{\left[ x + (y-x)\frac{k}{n},x + (y-x)\frac{k+1}{n}\right)}_{I_k}$$ Then $$\mu(I_k) \leq \nu(I_k)^a = \left(\frac{y-x}{n}\right)^a$$ Now by additivity we have $$\mu(I) = \sum_{k=0}^{n-1} \mu(I_k) \leq \sum_{k=0}^{n-1}\left(\frac{y-x}{n}\right)^a = (y-x)^a n^{1-a} \stackrel{n \to \infty}{\longrightarrow} 0 $$ Hence $\mu(I)=0$ for every interval so $\mu = 0$.

Answer 2

Hint: Notice that $[a,a+b)=[a,a+b/2)\cup [a+b/2,a+b)$. Use this to show that $\mu([a,a+b))$ is smaller than any $\varepsilon>0$.