Let A and B be probability measures on the probability space (X,Y) with pdfs f and g with respect to a measure $\displaystyle \mu $ on (X,Y). Show that $\displaystyle \int |f-g|d \mu = 2 \sup _{Z \in Y} |A(Z)-B(Z)| $
Let A and B be probability measures on the probability space (X,Y) with pdfs f and g with respect to a measure $\displaystyle \mu $ on (X,Y). Show that $\displaystyle \int |f-g|d \mu = 2 \sup _{Z \in Y} |A(Z)-B(Z)| $