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.



Are the partial derivatives of a function increasing in both variables measurable?
Yes. As Nik Weaver's comment points out, your function is automatically measurable. Therefore you can apply Theorem 1 of [Measurability of partial derivatives](http://www.ams.org/journals/proc/1977-063-02/S0002-9939-1977-0437696-1/) by Moshe and Mizel.


