Recover an $L^1$ Integrand by Partial Differentiation

Question

Denote by $m$ the 2-dimensional Lebesgue measure on $\mathbb{R}^2$. Let $f$ be an element of $L^1(m)$ that takes only nonnegative values. Define $F : \mathbb{R}^2 \longrightarrow [0,\infty)$ by $\displaystyle F(x,y) = \int\limits _{(-\infty,x] \times (-\infty,y]} fdm$, where $(x,y) \in \mathbb{R}^2$. Is there a counterexample to the statement that $\partial _2\partial _1F = f$ a.e.? Or this is actually a true statement?

Answer 1

Lemma 1 in Serrin J. (1961), On the Differentiability of Functions of Several Real Variables. Archive for Rational Mechanics and Analysis, 7, 359–372.