I will change it according to your suggestion.

@MaxAlekseyev: No, I am not considering multisets the elements of $X_N$ are pairs of distinct elements of $X_{N-1}$. It is only by looking into the nested sets of sets down to level $N = 0$ that one finds the elements of $X_0$. But maybe it is possible to reformulate the problem in terms of multisets. I will add a picture that should describe the problem more clearly.

I have no idea of the terms you are using. I am building these sets as example for a method for information theory that I am developing.

It is really a quite embarassing that I have not seen the signal processing inequality in my own question, right now I want to hide myself :-) Thanks for your time and your answer.