I have come across with the following question, find an example of a function $ F: ℝ^2 → [0, 1] $, such that : (i) $ x → F(x,y) $ is monotonically increasing for all $ y ∈ R $, (ii) $ x → F(x,y) $ is increasing for all $ x ∈ R $, (iii)there exists no measure $μ$ on $(R^2, B^2)$ with $ F (x, y) = μ((−∞, x] × (−∞, y]) $.
I am sure that in order to find this function i should think about the properties of Lebesgue-Borel measure or Lebesgue-Stieltjes measure but i am not quite sure and i have stuck in how i could combine all these 3 prerequisite. Any thoughts? Thanks in advance!!