# Lebesgue measurable set with measure $\overline{\lambda}^2(C_A)=2\pi\int_{A}t\overline{\lambda}(t)$

I'm trying to prove the next:

Let $$A\subset[0,\infty)$$ be a lebesgue measurable set and let $$C_{A}=\{(x,y)\in\mathbb{R}^{2}:\sqrt{x^2+y^2}\in A\}.$$

Prove:

i)$$C_A$$ is Lebesgue measurable.

ii)$$\overline{\lambda}^2(C_A)=2\pi\int_{A}td\overline{\lambda}(t)$$

iii) Generalize i) to other functions $$F:\mathbb{R}^{2}\rightarrow\mathbb{R}.$$

I've proved i) utilizing $$A$$ is Lebesgue measurable; then $$A$$ is union of a borelian set $$B$$ and a null Lebesgue set $$N$$. To prove this is enough to prove $$\overline{\lambda}^2(C_N)=0,$$ that is because $$N$$ is null set, and the Borel measurable follows from the continuous map $$F(x,y)=\sqrt{x^2+y^2}.$$

I'm having problems with ii). I've tried to use $$x-$$section of indicator function: $$1_{\{(C_A)_x\}},$$ but I don't know how to proceed to compute the integral. Here $$(C_A)_x=\{y\in\mathbb{R}:(x,y)\in C_A\}.$$

For iii) I suppose any measurable function $$F:\mathbb{R}^{2}\rightarrow\mathbb{R}$$ could work well.

Any kind of help is thanked in advanced.