How do I determine whether a function is a Borel funtion in multiple dimensions? (1D example solved here - but I can't seem to generalize it.)
Example
$$ v:\mathbb R^2 \rightarrow \mathbb R, ~~ v(x,y)=\text{ceil}(x^2 + y^3) $$
($\text{ceil}$ means round up to nearest integer), and we want to show that it is a Borel function ($\mathcal B(\mathbb R^2)/\mathcal B(\mathbb R)$)
I get a little confused once I get into higher dimensions - any help would be appreciated.
My thoughts
We want to find the preimage:
$$ v^{-1}((-\infty, a]) = \{ (x,y) : v(x,y) \leq a \} $$
From there I need to determine the values of that fit into the preimage (i.e. described as two range of values), which I would like to be an expression of $a$ (?). The set can be described as:
$$ \{x^2 + y^3 < \text{floor}(a) \} $$
Which can be described as
$$ (-\infty, x], (-\infty, (-x^2+a)^{1/3}] $$
This is an element of $\mathbb R^2$, whereby we conclude that $v$ is a Borel function.
I'm not sure how much measure theory you've done, but what copper.hat is suggesting is to use the fact that a composition of measurable functions is measurable. This reduces the problem to proving $$ \text{floor} : \mathbb R \to \mathbb R$$ and $$ f: \mathbb R^2 \to \mathbb R, \quad f(x, y) = x^2 + y^3 $$ are measurable (where we take the Borel $\sigma$-algebras throughout). $f$ is measurable since it is continuous and $\text{floor}$ is measurable since it is monotonic (you might want to prove this) so we're done.