Prove that this function is Borel Measurable.

Question

I have this function and I want to prove it is Borel measurable. $$\begin{equation} f:\mathbb{R}^2\rightarrow\mathbb{R}:f(x,y)=\begin{cases} \sin(\frac{1}{x-y}) &\text{ for } x>y \\\ x^2+y^2 &\text{ for } x\leq y \end{cases}\end{equation}$$ I thought you could start with only looking at the first part (so for $x>y$) and then the other part. Because if they were both Borel measurable then the function is too. But now I don't know how to do them apart.

Answer 1

Let $h(x,y)=\frac 1 {x-y}$ if $x >y$ and $0$ otherwise. Then $f(x,y)=\sin (h(x,y))I_A(x,y)+(x^{2}+y^{2})I_B$ where $A=\{(x,y): x>y\}$ and $B=\{(x,y): x\leq y\}$. Since products of Borel measurable functions, compositions of Borel measurable functions and sums of Borel measurable are Borel measurable it is enough it is enough to check that $h$ is Borel measurable. For this it is enough to check that $\{(x,y): h(x,y) <a\}$ is measurable for any $a \geq 0$ I will leave this last part to you.