# Relationship between deterministc function of random variables

Given a discrete $$P(X,Y,Z)$$ let's call $$\Omega$$ the set of all deterministic functions $$f: XYZ \rightarrow W$$ and $$\Omega'$$ the set of all deterministic functions $$f': XY \rightarrow V$$. Is it correct that $$\Omega' \subseteq \Omega$$? I am thinking that a deterministic function of a discrete random variable is essentially a partition of the outcomes of the random variable, thus the inclusion should be correct, but am not sure.

Technically, no, because the domain of the functions in the two sets is not the same. However, with a slight alteration, you can obtain the subset relation you want. If you let $$\Omega$$ be the class of all (measureable) functions of $$(X,Y,Z)$$ then you can define the subclass of functions:

$$\Omega' \equiv \{ f \in \Omega | f(x,y,z) = f(x,y,z') \text{ for all } z, z' \in \mathcal{Z} \},$$

where $$\mathcal{Z}$$ denotes the range of possible values of $$z$$. This gives you a subclass $$\Omega' \subseteq \Omega$$ containing functions that have the same domain as the functions in $$\Omega$$, but where all the functions in the subclass are invariant to the input value $$z$$.

(Incidentally, this question really has nothing to do with random variables --- it is just a question about functions. In the context of random variable inputs we would require the functions to be measureable, but that is the only difference.)

• (+1) for rigorous explanation. Jan 29, 2021 at 9:52

I think, in a more general sense, X,Y,Z don't have to be random variables let alone discrete. Intuitively, what you write resembles the following question:

Can we define all deterministic functions $$f(x,y)$$ in the form $$f(x,y,z)$$ instead?

This is yes. However, rigorously, the answer to your question is false because the domains of the functions are different.