Calculating Explicit Radon-Nikodym Derivatives I am stuck on the following problem:

Let $f(x,y) = \max \{ x^2 + y^2 , 1 \}$ and define a Borel measure $\mu$ on $\mathbb{R}$ by $\mu(E) : = (m \times m)(f^{-1}(E))$, where $m$ is Lebesgue measure. Find the Radon-Nikodym derivative $d \rho/d m$, where $\rho$ is the absolutely continuous part of $\mu$.

It is easy to see that $\mu$ is a positive measure (but not finite), $\mu(- \infty, 1)) = 0$, and $\mu$ is not absolutely continuous with respect to $m$ (for instance $m(\{ 1\}) = 0$ while $\mu(\{1\}) = \pi$). I feel like I have a solid understanding of the measure, specifically it is not too difficult to compute the measure of intervals, but I am unsure how to go about the actual problem. 
In Folland's analysis text, his proof of the Lebesgue Radon Nikodym Theorem is not constructive and did not help me out too much. It is clear that I am looking at this problem in the wrong way, so I was hoping for a helpful hint in the right direction. 
 A: According to Lebesgue's decomposition theorem, the measure $\mu$ decomposes as $$\mu = \rho + \nu,$$
 where 


*

*$\rho$ is absolutely continuous with respect to the Lebesgue measure $m$;

*$\nu$ and $m$ are singular with respect to each other.
Furthermore, this decomposition is unique.
This uniqueness of decomposition is key. It means that a possible way of solving this problem is to guess what $\rho$ and $\nu$ might be, based on intuition, then verify our the guesses obey the above properties.
As you say, it's not hard to come up with the right intuition here. To paraphrase your description:


*

*The "continuous" part of the measure is supported on $(1, \infty)$. By elementary calculus, we would expect that this continuous part is given by the 
$\rho(E) =  \int_{\mathbb R}  \pi .1_{E \cap (1,\infty)} \ dm$.

*The "discrete" part of the measure is supported on $\{ 1 \}$. It is simply $\nu(E) = \pi $ if $1 \in E$, or $\nu(E) = 0$ if $1 \notin E$.
So what needs to be done to formalise this?


*

*We need to show that my $\rho$ really is absolutely continuous w.r.t. $m$. This should be obvious, given the definition of $\rho$. It should also be obvious what the Radon-Nikodym derivative $d\rho / dm$ is.

*We need to show that $\nu$ and $m$ are singular. This shouldn't be too difficult either.

*We need to show that $\mu = \rho + \nu$. This can be done by switching to polar coordinates using the Jacobian change of variables formula...
