Definition
Let $F:\mathbb{R}^2\rightarrow \mathbb{R}$ be a function satisfying the following:
(i) $F(x,y)$ is right-continuous with respect to $x$
(ii) $F(x,y)$ is right-continuous with respect to $y$
Let’s say $F$ is a Stieltjes function in this case.
Let $F,G$ be two Stieltjes functions.
If $$F(a_1,b_1)-F(a_2,b_1)-F(a_1,b_2)+F(a_2,b_2)=G(a_1,b_1)-G(a_2,b_1)-G(a_1,b_2)+G(a_2,b_2)$$ for every $a_1,a_2,b_1,b_2$, then is $F-G$ constant?