Absolutely Continuous Measure and the Radon-Nikodym Derivative

Question

Let $\mu$ be a regular Borel Measure on $\mathbb{R}$ and suppose that there exists a $C > 0$ such that for every $x \in \mathbb{R}$ and $r > 0$ \begin{align*} \mu((x - r,x+r)) \leq Cr. \end{align*} Let m be the Lebesgue measure on $\mathbb{R}$. I have shown that $\mu << m$ ($\mu$ is absolutely continuous wrt m). I now want to show that the Radon-Nikodym derivative of $\mu$ wrt m is in $L^{\infty}(\mathbb{R},m)$.

I know that $\mu$ and m are two $\sigma$-finite measures and $\mu << m$. So, by the Radon-Nikdoym Theorem, there exists a positive non-negative measure function f so that \begin{align*} \mu(E) = \int_E f dm \end{align*} for every measurable set E. So, I need to show that $f \in L^{\infty}(\mathbb{R},m)$. So I need to show that for every $N \in \Sigma$ with $m(N) = 0$ there exists $C > 0$ so that \begin{align*} \left\vert{f(s)}\right\vert \leq C \end{align*} for every $s \in N^c$.

I am not sure how to proceed/how to relate this to the above integral or the first equation. Any ideas or recommendations would be greatly appreciated. Thank you in advance.

Answer 1

First you have $f \ge 0$, so you do not need to worry about taking absolute values.

Hints:

1. Can you show $\mu( A ) \le d \, m(A)$ for all $A$ and some constant $d > 0$?
2. Assume that $f \ge c$ on some (measurable) set $A$. What does this tell you about $\mu(A)$ and $m(A)$?