Why is $\{(x, y) \in [a, b] \times [a, b]: f(x) \geq y\}$ a Borel set? Suppose $f: I = [a, b] \rightarrow I$ is a Borel function. Why is $\{(x, y) \in [a, b] \times [a, b]: f(x) \geq y\}$ a Borel set?
Since $f$ is a Borel function, for any open set $U$, $f^{-1}(U)$ is Borel. However, I can't seem to be able to write $\{(x, y) \in [a, b] \times [a, b]: f(x) \geq y\}$ as the inverse image of an open set.

Let $f:I\rightarrow I$ be a Borel function. Let $F=\{(x,y)\in I\times I: f(x)\geq y\}$. Consider the map $g:I\times I\rightarrow I$ defined by $g(x,y)=f(x)-y$. If you can show that $g$ is Borel you'll be done, for then $F=g^{-1}([0,\infty))$. 
