Let $f$ be a function from $[0,1]\times[0,1]$ to $\mathbb{R}$ that is nondecreasing in both variables, i.e. $f(x_1,y_1)\le f(x_2,y_2)$ whenever $x_1\le x_2$ and $y_1\le y_2$. It is known that the partial derivatives $\partial f/\partial x$ and $\partial f/\partial y$ exist almost everywhere, but are they Lebesgue measurable?

The one-dimensional version of this problem is well-studied, and it is known that the restriction of $\partial f/\partial x$ to the set $[0,1]\times\{y\}$ is measurable for any $y$. It is also known that $\partial f/\partial x$ is measurable when restricted to the subset of $[0,1]\times[0,1]$ where $f$ is differentiable.