Let us say we define a coupling of two measures $\mu,\nu$ (reference : http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf). It seems that a coupling can't be defined if $\int d \mu \neq \int d \nu$. For example, it is said here in the definition of a Frechet class which seems to me to be the class of all couplings : https://warwick.ac.uk/fac/sci/statistics/staff/academic-research/kendall/personal/talks/CouplingReview-handout-2x2.pdf
my question is why so?
The answer comes down to the definition of a coupling. One wondering the answer should look up the definition of product space, marginals and projection. Also, intuitively, how would one couple two measures with different masses?