I am familiarizing myself with concepts of causality by working through the book Elements of Causal Inference by Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. They state the following problem (Problem 3.8b):
Consider the cyclic structural causal model (SCM): $$X:=Y+N_X$$ $$Y:=X+N_Y$$ with $(N_X, N_Y) \sim P$. Show that if $P$ allows for a density with respect to Lebesgue measure and factorizes, that is, $N_X ⊥ N_Y$, then there is no solution $(X, Y, N_X, N_Y)$ of the SCM.
We understand $:=$ as an assignment operation. For a solution to exist, we would have to have $N_X = -N_Y$, which already violates the independence assumption. Why do we need the density with respect to the Lebesgue measure?
