Quantifying over all random variables I often encounter statements in the literature in probability theory of the form:
"Let $(\Omega, \mathscr{A}, P)$ be a probability space, $S$ a state space and $X : \Omega \to S$ a random variable with some properties ... .
Then there is a random variable $Y$ equivalent to $X$ in distribution if and only if the probabilistic property $\phi$ of $X$ is true."
This statement says that one can choose a random variable $Y$ with some special functional properties beside the purely probabilistic properties given by the distribution of $X$ (resp. $Y$).
The problem here is, that in this statement, there is a quantification over all random variables, which to my opinion do not form a set. The underlying probability space for the choice of $Y$ is not specified. One can only speak of equivalence in distribution if two random variables $X$ and $Y$ together with their probability spaces $(\Omega, \mathscr{A}, P)$ and $(\Omega', \mathscr{A}', P')$ are already given (wherever they come from) and it holds that $P \circ X^{-1} = P' \circ Y^{-1}$. In fact, in order to be precise, a random variable $X$ is the complete tuple $(\Omega, \mathscr{A}, P, X)$ and not only the function $X : \Omega \to \mathbb{R}$.
So, how should those statements be interpreted?
I think, the meaning of these statements is that the definition of the random variable $Y$ with the desired properties can be done explicitely. So, the statement should read as:
"Let $(\Omega, \mathscr{A}, P)$ be a probability space and $X : \Omega \to \mathbb{R}$ a random variable with some properties ... .
(1) If $X$ satisfies $\phi$ then define $(\Omega', \mathscr{A}', P')$ by ... and $Y : \Omega' \to S$ by ...  and it holds that $Y$ is equivalent in distribution to $X$.
(2) Let $(\Omega', \mathscr{A}', P')$ be given and $Y : \Omega' \to S$ a random variable equivalent in distribution to $X$. Then $X$ satisfies some probabilistic property (which in turn is also satisfied by $Y$ since the property is of probabilistic nature and thus based on the distribution).
How would you interprete such statements?
 A: Tis true, that the collection of all random variables is not a set. Because the collection of "state spaces" is not a set.
But this is now a statement in the language of set theory. It's a quantification over the universe of sets, and now it's fine. Like saying that "For every set $x$, $x\notin y$" or "For every set $x$, $|x|<|\mathcal P(x)|$" are quantifications over all sets.
The statement, when translated to the language of set theory says that whenever you have a set which satisfies the properties of being a state space, and so on and so forth, is a very complicated statement in the language of set theory, but it can be written nonetheless.
(Here is a similar example, taken from a course I am currently attending: If $X$ is a Polish space, $A\subseteq X$ is Universally Baire if for every topological space $Y$ and a continuous function $f\colon Y\to X$, $f^{-1}(A)$ has the Baire property in $Y$. Here we quantify over all topological spaces and continuous functions into $X$, and that's a legitimate statement about sets, so we can do it as a statement about the universe of sets.)
