Consider the SVD of rectangular matrices as operators
$$ svd : \mathbb C^{n\times m} \to \mathbb U_n \times \mathbb D_{n,m}\times \mathbb U_m $$
where $\mathbb U$ is the space of $n\times n$ unitary matrices and $\mathbb D_{n,m}$ is the space of real nonnegative diagonal rectangular matrices.
We know that this operator is not well defined, since there's more than one SVD for each matrix.
What I want to know is, there exists a specification of $svd$ on $\mathbb C^{n\times m}$ that makes it a Borel-measurable function?
I already know that it can't be continuous, and that if I play enough with the signs of the entries, I guess I can make it not even Lebesgue measurable, but what I need here is for just one specification. If I had to try to do it, I guess I'd try to define it locally and then try to glue together the charts, but I feel like it might badly backfire.
