I'm trying to prove the next:

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


i)$C_A$ is Lebesgue measurable.


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.


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